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TRANSPORTATION AND TRAFFIC THEORY FLOW, DYNAMICS AND HUMAN INTERACTION Proceedings of the 16th International Symposium on Transportation and Traffic Theory University of Maryland, College Park, Maryland, 19-21 July 2005
edited by
HANI S. MAHMASSANI University of Maryland, USA
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Contents Preface
ix
International Advisory Committee
xi
Contributors
xii
Bilevel optimisation of prices in a variety of transportation models
1
MJ Smith A sequential experimental approach for analyzing second-best road pricing with unknown demand functions H Yang, WXu and Q Meng
23
Highway space inventory control system D Teodorovic and P Edara
43
Path size and overlap in multi-modal transport networks SHoogendoorn-Lanser, R van Nes andP HL Bovy
63
Pedestrian dynamics and evacuation: empirical results and design solutions D Helbing, A Johanssen and L Buzna
85
An empircal assessment of traffic operations C Chen, P Varaiya and J Kwon
105
Reliability of freeway traffic flow: a stochastic concept of capacity WBrilon , J Geistefeldt and M Regler
125
A critical comparison of the kinematic-wave model with observational data K Nagel and P Nelson
145
Average velocity of waves propagating through congested freeway traffic B Coifman and Y Wang
165
Microscopic three-phase traffic theory and its applications for freeway traffic control B Kerner
181
A behavioural approach to instability, stop and go waves, wide jams and capacity drop C Tampere, S Hoogendoorn and B van Arem
205
Controlling traffic breakdowns R Ktihne and R Mahnke
229
vi Contents Parameter estimation and analysis of car-following models S Hoogendoorn and S Ossen
245
Modeling Impatience of drivers in passing maneuvers M Pollatschek and A Polus
267
A simulation model for motorway merging behaviour J Wang, R Liu and F Montgomery
281
Freeway ramp merging process observed in congested traffic: lag vehicle acceleration model M Sarvi, A Ceder and M Kuwahara
303
A first-order macroscopic traffic flow model for mixed traffic including moving bottleneck effects S Chanut
323
A variational formulation of kinematic waves: bottleneck properties and examples C Daganzo and M Menendez
345
First-order macroscopic traffic flow models: intersection modeling, network modeling J P Lebacque and M Khoshyaran
365
Real-time estimation of travel times on signalized arterials A Skabardonis and N Geroliminis
387
Calibration and validation of dynamic traffic assignment systems R Balakrishna, H Koutsopoulos and M Ben-Akiva
407
Non-equilibrium dynamic traffic assignment W Y Szeto and H K Lo
427
Precision of predicted travel time, the responses of travellers, and satisfaction in the travel experience SKikuchi, S Mangalpally and A Gupta
447
Behavioral dynamics in activity participation, travel, and information and communications technology K Goulias and T G Kim
467
Modeling the joint labor-commute engagement decisions of San Francisco Bay Area residents
487
DTOry and P L Mokhtarian
Contents vii A model of daily time use allocation using fractional logit methodology X Ye and R Pendyala
507
Efficient estimation of nested logit models using choice-based samples L A Garrow, F S Koppelman and B L Nelson
525
Functional approximations to alternative-specific constants in time-period choice-modelling S Hess, J W Polak and M Bierlaire
545
Project-based activity scheduling for person agents
565
EJ Miller Modelling commercial vehicle empty trips: theory and application JHolguin-Veras, JCZorrilla andE Thorson
585
Rationality and heterogeneity in taxi driver decisions: application of a stochastic-process model of taxi behavior R Kitamura and T Yoshii
609
A rolling-horizon approach to the optimal dispatching of taxis
629
M GHBell, KI Wong and A J Nicholson Capacitated arc routing problem with extensions H Qiao and A Haghani
649
User-equilibrium route set analysis of a large road network H Bar-Gera and D Boyce
673
Comparison of static maximum likelihood origin-destination formulations H Rakha, H Paramahamsan and M van Aerde
693
A time-dependent activity and travel choice model with multiple parking options 717
HJHuang, Z CLi, WHKLam andS C Wong Doubly dynamic equilibrium distribution approximation model for dynamic traffic assignment N C Balijepalli and D P Watling
741
A combined model of housing location and traffic equilibrium problems in a continuous transportation system H W Ho and S C Wong
761
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Preface For many researchers and scholars in the inter-disciplinary field of traffic science and transportation system modelling, the book publication that contains the Proceedings of the International Symposium on Transportation and Traffic Theory (ISTTT) series provides a much-awaited field-defining milestone, featuring some of the best and most promising thinking and theoretical developments in the field. The ISTTT series is the main gathering for the world's transportation and traffic theorists, and those who are interested in contributing to or gaining a deep understanding of traffic and transportation phenomena in order to develop effective approaches to plan, design and manage these systems. The Symposium deals primarily with the scientific and fundamental aspects of transportation and traffic phenomena, including identification and characterization of previously undocumented or underappreciated phenomena, formulation, observation and testing of new models of and new perspectives on known phenomena, as well as extension, refinement, testing or critique of existing theories and models of known phenomena. This volume contains contributions selected for the 16th International Symposium on Transportation and Traffic Theory (ISTTT), held at the University of Maryland in College Park, Maryland in the USA on July 19 - 21, 2005. The 39 contributions selected for this volume do not represent all the papers accepted for inclusion in the ISTTT program. In view of the high level of interest and very high quality of submissions from all over the world, and the limited number that could be included in this volume, several additional papers were accepted for following a very demanding and rigorous refereeing process. These will be published through regular Journal channels. All papers in this volume passed a three-stage review process. In a first stage, nearly 200 submitted extended abstracts were reviewed by up to four members of the scientific and organizing committees, resulting in invitations to 88 authors to submit full papers. The second stage entailed the review of each of the 84 submitted full papers by two to five internationally recognized referees. Based on that process, 45 revised papers were submitted for final evaluation and review; 39 of these are included in the present volume, each reflecting the high quality, excellence, rigour and innovation that are the defining qualities of this series. The ISTTT 16 comes at a time when the field is experiencing a major renewal, reflected by the entry of a large number of new scientists from a variety of disciplines, and the mutual coexistence of a growing number of theoretical perspectives and modelling cultures. It is also a time when theories and models developed over the past 40 years are finding their way into professional practice through computer software packages to a greater extent than at any time in the past. The present volume reflects some of the most vibrant work taking place in the world's leading institutes on the science of traffic and transportation, and presents a view of the breadth of topics and richness of influences that are permeating the field.
x Preface The contributions included in this volume reflect a range of topics that pertain to modelling traffic and transportation processes as complex systems. Several themes are reflected in this collection, though perhaps none as strongly and visibly as the overarching desire to understand and characterise the fundamental processes that govern the behaviour and performance of our transportation systems. At the root of these characterizations is the recognition that traffic and transportation systems are a collective expression of individual human decisions. Modelling traffic and transportation systems is about modelling human interaction, amongst individuals as well as with the surrounding transportation infrastructure, controls and environment. The number of contributions in this volume that deal with some aspect of human behaviour, whether as travellers, drivers, passengers, operators, or regulators, is relatively greater than in past Symposia. This is reflective of great strides being made in our community of scholars in developing theories and mathematical representations of these phenomena, and in the combination of theoretical development with observation through novel measurement techniques. The underlying themes and approaches traditionally used in different functional areas, such as planning or operations, or transport modes, such as freight or passenger modes, appear to be converging towards greater focus on individual decisionmaking and human interaction in the context of flow processes, network dynamics and spatial systems. Special appreciation is extended to the members of the International Advisory Committee and the local organizing committee in the review and selection process of the papers included in this volume. Particular gratitude is owed to the referees who contributed considerable time and effort to the review process. Only through their dedicated and thoughtful effort can the ISTTT maintain its high standards of intellectual excellence. I am grateful to Ms. Elisabeth Fernandez-Kimmel for her most able assistance and tremendous effort in the final preparation of the camera-ready document. Hani S. Mahmassani March 2005
International Advisory Committee C F Daganzo
University of California, Berkeley, USA (Convenor)
R E Allsop
University College, London, UK
M G H Bell
Imperial College, London, UK
P H L Bovy
Delft University of Technology, The Netherlands
W Brilon
Ruhr-University, Bochum, Germany
A Ceder
Technion-Israel Institute of Technology, Haifa, Israel
N H Gartner
University of Massachusetts, Lowell, USA
R Kitamura
Kyoto University, Japan
R Kiihne
German Aerospace Center, Berlin, Germany
M Kuwahara
University of Tokyo, Japan
WLam
Hong Kong Polytechnic University, China
J-B Lesort
Institut National de Recherche sur les Transports et leur Securite, Lyon, France
H.S. Mahmassani
University of Maryland, College Park, USA
V V Silyanov
Moscow State Automobile and Road Technical University, Russia
M A P Taylor
University of South Australia, Adelaide, Australia
M Tracz
Cracow University of Technology, Poland
S C Wirasinghe
University of Calgary, Canada
Honorary members E Hauer
Canada
H Keller
Germany
M Koshi
Japan
W Leutzbach
Germany
H-G Retzko
Germany
D I Robertson
United Kingdom
S Yagar
Canada
Contributors R Balakrishna N C Balijepalli H Bar-Gera M G H Bell M Ben-Akiva M Bierlalre P H L Bovy D Boyce W Brilon L Buzna A Ceder S Chanut CChen B Coifman
C Daganzo P Edara
L A Garrow J Geistefeldt N Geroliminis K Goulias A Gupta A Haghani D Helbing SHess HWHo
Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Institute for Transport Studies, University of Leeds, UK Ben-Gurion University of the Negev, Be'er Sheva, Israel Centre for Transport Studies, Imperial College London, UK Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Institute of Mathematics, Ecole Polytechnique Federate de Lausanne, Switzerland Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft University of Technology, The Netherlands Northwestern University Evanston, Illinois, USA Institute for Transportation and Traffic Engineering, RuhrUniversity, Bochum, Germany Department of Transportation Networks, University of Zilina, Slovakia Transportation Research Institute, Technion-Israel Institute of Technology, Haifa, Israel Laboratoire Ingenierie Circulation Transport LICIT (INRETS - ENTPE), Cedex, France University of California, Berkeley, USA Department of Civil and Environmental Engineering and Geodetic Science, Ohio State University, Columbus, Ohio, USA Institute of Transportation Studies, University of California, Berkeley, USA Charles E. Via Jr. Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Falls Church, Virginia, USA Georgia Institute of Technology, Atlanta, Georgia, USA Institute for Transportation and Traffic Engineering, RuhrUniversity, Bochum, Germany Institute of Transportation Studies, University of California, Berkeley, USA University of California, Santa Barbara, USA University of Delaware, Newark, Delaware, USA Department of Civil and Environmental Engineering, University of Maryland, College Park, Maryland, USA Institute for Economics and Traffic, Dresden University of Technology, Dresden, Germany Centre for Transport Studies, Imperial College London, UK Department of Civil Engineering, The University of Hong Kong, China
Contributors
HWHo J Holguin-Veras S Hoogendoorn S Hoogendoorn-Lancer H J Huang A Johanssen B Kerner M Khoshyaran S Kikuchi TGKim R Kitamura
F S Koppelman H Koutsopoulos R Kiihne M Kuwahara JKwon W H K Lam J P Lebacque ZCLi RLiu HKLo R Mahnke S Mangalpally M Menendez Q Meng E J Miller P L Mokhtarian
xiii
Department of Civil Engineering, The University of Hong Kong, China Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft University of Technology, The Netherlands Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft University of Technology, The Netherlands School of Management, Beijing University of Aeronautics and Astronautics, China Department of Physical Resource Theory, Chalmers University of Technology, Sweden Daimler Chrysler AG, Stuttgart, Germany ETC Economics Traffic Clinic, France University of Delaware, Newark, Delaware, USA The Pennsylvania State University, University Park, Pennsylvania, USA Department of Urban Management, Kyoto University, Kyoto, Japan, and Department of Civil and Environmental Engineering, University of California, Davis, USA Northwestern University Evanston, Illinois, USA Department of Civil Engineering, Northeastern University, Boston, Massachusetts, USA Institute of Transportation Research, German Aerospace Center, Berlin, Germany Institute of Industrial Science, University of Tokyo, Japan Statistics Department, California State University, Hayward, California, USA Department of Civil and Structural Engineering, Hong Kong Polytechnic University, China INRETS-GRETIA2, France School of Management, Beijing University of Aeronautics and Astronautics, China Institute for Transport Studies, University of Leeds, UK Department of Civil Engineering, Hong Kong University of Science and Technology, Hong Kong, China Uiversity of Rostock, Institute of Physics, Rostock, Germany University of Delaware, Newark, Delaware, USA Institute of Transportation Studies, University of California, Berkeley, USA Department of Civil Engineering, The National University of Singapore, Singapore Department of Civil Engineering, University of Toronto, Toronto, Canada University of California at Davis, Davis, USA
xiv Contributors F Montgomery K Nagel B L Nelson P Nelson A J Nicholson DTOry S Ossen H Paramahamsan R Pendyala J W Polak M Pollatschek A Polus HQiao HRakha M Regler M Sarvi A Skabardonis M J Smith W Y Szeto C Tampere D Teodorovic
E Thorson M van Aerde B van Arem R van Nes P Varaiya
Institute for Transport Studies, University of Leeds, UK Institute for Land and Sea Transport Systems, Technical University of Berlin, Berlin, Germany Northwestern University Evanston, Illinois, USA Dept of Computer Science, Texas A&M University, College Station, Texas, USA Deparment of Civil Engineering, University of Canterbury, New Zealand University of California at Davis, Davis, USA Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft University of Technology, The Netherlands Charles Via Jr. Dept. of Civil and Environmental Engineering, Virginia Tech, Blacksburg, Virginia, USA Department of Civil and Environmental Engineering, University of South Florida, Tampa, Florida, USA Centre for Transport Studies, Imperial College London, UK Department of Industrial Engineering and Management, Technion-Israel Institute of Technology, Haifa, Israel Department of Civil and environmental Engineering, Technion-Israel Institute of Technology, Haifa, Israel The Robert H. Smith School of Business, University of Maryland, College Park, Maryland, USA Charles Via Jr. Dept. of Civil and Environmental Engineering, Virginia Tech, Blacksburg, Virginia, USA Institute for Transportation and Traffic Engineering, RuhrUniversity, Bochum, Germany Department of Civil Engineering, Monash University, Victoria, Australia Institute of Transportation Studies, University of California, Berkeley, USA Department of Mathematics, University of York, UK Department of Civil, Structural, and Environmental Engineering, Trinity College, University of Dublin, Ireland Department of Traffic and Infrastructure, Katholieke Universiteit, Leuven, Belgium Charles E. Via Jr. Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Falls Church, Virginia, USA Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA Posthumously TNO Inro, Department of Traffic and Transport and University of Twente Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft University of Technology, The Netherlands University of California, Berkeley, USA
Contributors xv J Wang Y Wang D P Watling K I Wong S C Wong W Xu H Yang X Ye T Yoshii J C Zorrilla
Institute for Transport Studies, University of Leeds, UK Department of Electrical and Computer Engineering, Ohio State University, Columbus, Ohio, USA Institute for Transport Studies, University of Leeds, UK Centre for Transport Studies, Imperial College London, UK Department of Civil Engineering, The University of Hong Kong, China Department of Civil Engineering, The Hong Kong University of Science & Technology, Hong Kong, China Department of Civil Engineering, The Hong Kong University of Science & Technology, Hong Kong, China Department of Civil and Environmental Engineering, University of South Florida, Tampa, Florida, USA Department of Urban Management, Kyoto University, Kyoto, Japan Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA
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1
BILEVEL OPTIMISATION OF PRICES IN A VARIETY OF TRANSPORTATION MODELS MJ Smith, Department of Mathematics, University of York, United Kingdom
INTRODUCTION The primary equilibrium model in this paper is similar to that suggested by Aashtiani and Magnanti (1983). This model combines the standard "user equilibrium" route-choice principle stated by Wardrop (1952) and a demand for travel between each OD pair which may vary with the costs of travel between the various OD pairs. There are many other equilibrium models. See for example: Beckmann (1956), Evans (1976), Gartner (1980), Bar Gera (2002) and Bar Gera and Boyce (2003). Optimising prices and signal controls within equilibrium models The need to optimise controls within equilibrium models is clear; as Beckmann often pointed out, control changes have a knock-on effect as users play their own games while the controller is playing his. The basic optimisation idea in this paper is intended to be a really quite simple approach to the problem; building directly upon the "half-space projection" direction developed in a series of papers: Smith et al (1997, 1998a, b), Clegg et al (2001) and Clegg and Smith (1998, 2001). We extend this previous work and also show how these methods apply to a range of different models. Fletcher and Leyffer (2000) and Rodrigues and Monteiro (2004) both consider descent methods which have some similarity with the simultaneous descent method outlined here. Using the simultaneous descent algorithm here the approach to a stationary point is typically via points which are not themselves approximate equilibria, so that the optimisation and the equilibration move in parallel and the need to compute a sequence of approximate equilibria is avoided.
2 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Abdulaal and Leblanc (1979), Tan, Gershwin and Athans (1979), Marcotte (1983, 1986), Fisk (1984), Tobin and Friesz (1988), Davis (1994), Yang and Yagar (1994), Yang (1996a, b, c), Chiou (1997), Clune et al (1999), Patriksson and Rockafellar (2002) and Clark and Watling (2002) have all sought to solve bilevel problems. Migdalas (1995) provides an excellent survey of the area. Bilevel problems in a wider context have been studied by for example Gauvin and Savard (1994), Luo et al (1996) and Outrata and Zowe (1998). The need to optimise signal controls subject to equilibrium within an equilibrium transportation model was pointed out in an early paper by Allsop (1974). In this paper we begin by considering optimisation within our primary model and then show that the same ideas work in several other models. Finally we give a general model embracing all of them. THE BASIC EQUILIBRIUM MODEL, MODEL 0 The model, model 0, presented here is the base or initial model considered in this paper. The main variables In model 0 we suppose given a network with K OD pairs, that OD pair ij is joined by Ny routes and so the total number of routes is N = SijNij. The main variables are as follows: Xjjr = the flow along the r* route joining OD pair ij (in vehicles per minute, say); X = the route flow vector comprising all the Xyr; Yy = the cost of travel between OD pair ij (in minutes per vehicle, say); and Y = the cost vector comprising all the Yy. Flows are in (flow per unit time) units and costs are in (time per unit of flow) units; so that a flow times a cost is dimensionless. Let 0N denote the zero N-vector, +coN denote the N-vector with infinite co-ordinates. Then we put [0N, +ooN) = [0, +oo)N and [0K, +ooK) = [0, +oo)K. Cost functions and demand functions We suppose given two functions; the cost function C(.) and the demand function D(.). Here Cjjr(X) is the cost of traversing the rth route joining OD pair ij when the flow vector is X > 0 and Dy(Y) is the total flow between OD pair ij when the OD cost vector is Y where Y > 0. We suppose that the cost function C(.) is defined throughout [0N, +ooN) and that the demand function D(.) is defined throughout [0K, +ooK). Thus including domains and co-domains, our two given functions are: C: [0N, +ooN) -^ [0, +ooN) and D: [0K, +ooK) [0K, +ooK). The functions T and S We define the two functions T:[0N, +<x>N) * [0K, +ooK) and S:[0K, +°oK) -> [0N, +°oN) as follows. For each ij and each ijr: Tij(X) = ZrXijr for all X e [0N, +ooN) and S1]r(Y) = Yy for all Y e [0K, +ooK). Tjj(X) is the total flow from i to j and Syr(Y) spreads each cost Yy over all routes joining i to j .
Bilevel optimisation of prices in a variety of transportation models 3 Assumptions: positivity, non-negativity, boundedness, monotonicity, smoothness We suppose that: C(X) > 0 for all X e [0N, +ooN) and D(Y) > 0 for all Ye [0K, +ooK). D is bounded and C is bounded on bounded sets. C and -D are both monotone. C and D are differentiable and their derivatives are continuous. C is monotone means that: [C(X') - C(X2)]-(X'- X2) > 0 for all X1 e [0N, +ooN) and X2 e [0N, +ooN). It is easy to show that if C and - D are monotone then (C(X) - S(Y), T(X) - D(Y)) is also a monotone function of (X, Y) e [0N, +ooN) x [0K, +ooK).
EQUILIBRIUM CONDITIONS AND SEARCH DIRECTION Suppose given a (flow-vector, cost-vector) pair (X, Y) in which all Xyr > 0 and all Yjj > 0. Then (X, Y) will be called an equilibrium if (X, Y) s [0N, +ooN) x [0K, +ooK), for each ijr the cost Cijr(X) of traversing the rth joining OD pair ij equals Yjj, and for each ij the demand Djj(Y) generated by the cost vector Y equals the total flow Tjj(X) = ZrXjjr actually occurring from i to j . So in this case (X, Y) is a variable demand equilibrium pair if and only if (X, Y) e [0N, +coN) x[0K,+ooK), Yjj - Cijr(X) = 0 for all ijr; and Dij(Y) - Tjj(X) = 0 for all ij. (1) The equilibrium equations (1) may be written in vector form: (S(Y) - C(X), D(Y) - T(X)) = 0. Here we are assuming initially that at equilibrium all Xjjr > 0 and all Yjj > 0. To relax this initial assumption and so to allow the possibility that a listed route may have a high cost and zero flow at equilibrium we revise (1) to introduce a partial complementarity condition. The revised equilibrium condition is as follows. For each ijr: Yy - Cijr(X) < 0 and Yy - Cijr(X) < 0 implies Xijr = 0; and Dij(Y)-Tij(X) = 0. (2) The set of equilibria, or solutions to (2), will be denoted by E. Throughout this paper we assume that this set is non-empty. The feasible set F and the objective function V Suppose given upper bounds ViUXy,- and !4UYy for Xyr and Yjj at equilibria; so that all equilibria lie inside ViF and so well inside F where F = [0N, UX] x [0K, UY] = n ijr [0, UXij,] x riijtO, UYy] c [0N, +roN) x [0, +ooK). This ensures that the set of equilibria E is well inside F. We do not here consider the problem of proving existence of a non-empty set of equilibria, and nor do we consider the specification of the set F. Some results concerning the existence of UXjjr, UY;J and F are given in Smith (2005). Given the set F, consider the objective function V: F -> R+ where, for all (X, Y) e F:
4 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Ljr{Xijr2[Cijr(X) - Y y ] + 2 + (UXijr- XijO^Yij - Cijr(X)]+2} +Lj{Yij2[Tij(X) - Dij(Y)]+2+ (UYrYij)2[Dij(Y) - T,j(X)]+2}. Here y+ = max{y, 0} for all real numbers y. It is easy to show that, provided the set of all equilibria is a subset of !4F then the set of zeros of V in F is exactly the set of equilibria.
V(X, Y) =
To seek a zero of V and hence an equilibrium state, consider the following search direction A = (A1, A2) where the ijrth component of the flow search direction A'(X, Y) is for each ijr: A'i]r(X, Y) = (UXijr- Xijr)2[Y;j - Cijr(X)]+ - Xijr2[Cijr(X) - Y S ] + ; and the ijth component of the cost search direction A2(X, Y) is for each ij: A2ij(X,Y) = (UYi]-Yij)2[Dij(Y)-Tij(X)]+-Yij2[Tij(X)-Dij(Y)]+. The whole search direction A(X, Y) = (A'(X, Y), A2(X, Y)) for all (X, Y) e F. This is precisely the algorithm (D) direction, introduced in Smith (1984a, b), in this setting; applied with the function - (C(X) - S(Y), T(X) - D(Y)) on the set F = [0N, UX] x [0K, UY]. We are given that C and -D are both monotone and it follows that (C(X) - S(Y), T(X) - D(Y)) is a monotone function of (X, Y) on F. This is the fact which is critical to what follows. Since (C(X) - S(Y), T(X) - D(Y)) is a monotone continuously differentiable function of (X, Y) on F, it follows from Smith (1984a, b) that A(X, Y) is a descent direction for objective V at (X, Y) (unless V(X, Y) = 0 in which case (X, Y) is an equilibrium) and also that there is for each (X, Y) a positive number T\ such that (X, Y) + tA(X, Y) e F if (X, Y) e F and 0 < t < n. The set E of equilibria is under the present conditions also convex, since (C - S, T - D) is monotone. Since V is continuous the set E = {x; V(x) = 0} is also closed. Thus since we are assuming that E is non-empty, E is a closed convex set such that E c [0N, V2UX]x[0K, ViUY].
AN EQUILIBRATION METHOD Motivated by the previous section, we suppose that the following basic condition holds. BASIC CONDITION: There is a fixed set of N routes joining K OD pairs as above, C and -D are monotone and continuously differentiable, F = [0N, UX]x[0K, UY], and the set of equilibria is a non-empty subset of >/2F = [0N, y2UX]x[0K, '/2UY]. Any pair (X, Y) in F will be called feasible. In order to estimate an equilibrium (X, Y), we start anywhere in F and update (X, Y) as follows: (X1, Y1) is any feasible starting value for (X, Y) and following some rule or algorithm: (X1, Y1) ^ (X2, Y2) * (X3, Y3) ^ . For any feasible start point all succeeding pairs are also to be feasible.
Bilevel optimisation of prices in a variety of transportation models 5 Definition of convergence We make the following definition. The above infinite sequence {(Xn, Yn)} of feasible (X, Y) pairs in F converges if and only if (1) the equilibrium set E is nonempty and (2) the Euclidean distance between (X", Y") and the set E of equilibria tends to zero as n tends to infinity. We will agree that an algorithm converges if and only if for any start point (X1, Y1) which generates an infinite sequence then this infinite sequence converges. Steepness of descent Suppose now that our basic condition above holds. Then Smith (1984a, b) shows that (given the basic condition above) there is a continuous real-valued function G such that G(X,Y)>Oif(X,Y)eF\E; G(X, Y) = 0 if (X, Y) e E; and A(X, Y)-gradV(X, Y) < - G(X, Y) for all (X, Y) e F. This means that V is a Lyapunov function for the dynamical system: d((X(t), Y(t))/dt = A((X(t), Y(t)) for all t > 0, (X(0), Y(0)) = (X°, Y°) e F. It also means that the steepness of the descent of the Lyapunov function V at (X, Y), as (X, Y) follows A(X, Y), maybe estimated by G(X, Y). That is: dV(X(t), Y(t))/dt = gradV(X(t), Y(t)) A(X(t), Y(t)) < - G(X(t), Y(t)) for all t > 0. The projection Proj Given F, for each (X, Y) e RN+1C the projection Proj(X, Y) of (X, Y) onto the feasible set F is defined to be the point of F closest to (X, Y).
DYNAMIC ARMIJO-LIKE STEP LENGTHS Given that the basic condition holds we reduce V to zero by moving (X, Y) e F continually in direction A. For step length choice we follow a dynamic Armijo-like scheme very close to that described in Smith (1984b). For now we let z stand for (X, Y) and Zn stand for (Xn, Yn). A (zn, tn_i)-updating dynamic Armijo-like equilibration algorithm Here we suppose that if we are at iteration n; at a non-equilibrium (Xn, Y") = z,, where the search direction is A(Zn) and the step length actually used at Zn is un then our next z will be Zn+i = PrOJ(Zn + UnA(Zn))
where Proj denotes the projection onto F. The real number un > 0 will be called a used step length and tn > 0 will be called a step length. For the purposes of most of this section we will suppose that if u^ is a used step length then Proj(zn + UnA(z^)) = Zn + UnA(Zn).
6 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction That is we suppose that the projection operator does not actually do anything; so the boundary of the feasible set F is here having no effect. To determine un we specify a dynamic Armijo-like scheme based on a continuous function G such as that given above. To motivate the scheme, it is clear that if the step length t at z were very small then the change in V = V(z + tA(z)) - V(z) would be more negative than - 3/4tG(z). So the slope of V(z + tA(z)) against t would be at least as steep as - 3AG(z) for small t. We do not wish to have such small steps t as the reduction in V might be very small, by virtue of the small step size t. On the other hand if t is large the slope of V(z + tA(z)) against t can be no steeper than -%G(z) on average; for if the average gradient of V(z + tA(z)) against t were < - %G(z) for large t then V(z + tA(z)) would sometimes be < V(z) - VitG(z) and this would be negative for large t, which is impossible. (Of course, by its definition, V > 0 always.) We do not wish to have steps this large as the reduction in V might be very small, due to a shallow negative slope (or perhaps V may even increase, due to a positive slope). So we seek step lengths which give rise to slopes between - 3/4G(z) and - lAG{z). Such step lengths have an Armijo property and allow convergence to equilibrium to be shown. The dynamic Armijo (zn, tn_i)-updating equilibration algorithm To be specific, we start at an arbitrary z\ e F and to = 1. This 0th or initial step length to is fairly arbitrary, but it must be positive. If we are at a current non-equilibrium point Z j e F , and the previous possible step length was tn_i, then we are to update Zn and tn-i (for n > 1) according to some fairly simple rules as follows. Firstly, zn is kept fixed and tn.i is halved to obtain: tn_i, (i4)tn.i, (/4)2tn-i, . . . . , (V2)ptn-i, where the halving ceases as soon as: V[z^ + (MOViAfe)] - V(zn) < - (l/8)(I/2)ptn.,G(zn) for the first time, p = 0 is allowed here; it may be that Vfc, + t^Afe)] - V(Zn) < - (l/8)tn.1G(Zn) already. The halving surely ceases by definition of G as Zn is a non-equilibrium. Then: let un = (Vy'tn-i for this p (this is the used step length at zn) and let Zn+i = Zn + UnA(Zn).
Finally update tn_i as follows: if V f e + unA(zn)] - V(zn) < -3/4UnG(Zn) put tn = 2u n ; if -3/4UnG(zn) < V[zn + unA(zn)] - V(Zn) < -1/4UnG(zn), put tn = u n ; and i f - V ^ G f e ) < V[z» + unA(zn)] - V(zn) < - (l/8)u n G(z n ) put tn ='/2un. These are three mutually exclusive possibilities and they together exhaust all eventualities since, by choice of un = ('/iytn-ijVtzn + unA(Zn)] - V(z,,) > - (1/8)^0(2,,) is not possible.
Clearly not all tn are in fact used to move z. The used step lengths are the un. It is clear that V[zn + unA(zn)] - V(zn) < - (l/SKGfe) for all used step lengths un. (Stop when V(zT1) is less than some preassigned positive number.)
Bilevel optimisation of prices in a variety of transportation models 7
CONVERGENCE TO EQUILIBRIUM Suppose that the basic condition holds. For any starting point zi e F and with an initial "previous" step length to = 1 we suppose that the algorithm specified above generates an infinite trajectory: z\, Zi, Z3,.. ., z,,, . . . . and an infinite sequence to, ti, Xi, \-$,. .., tn, . . . . of possible step lengths. We assume that these sequences are infinite: so we never actually hit the equilibrium set E. (If the sequence hits an equilibrium we simply stop.) To prove convergence we use proof by contradiction. Suppose that {z,^ tn-i} is an infinite sequence generated by our algorithm and that {zj does not converge to the equilibrium set E = {z e F; V(z) = 0}. Then, since F is closed and bounded, {Zn} must have a non-equilibrium limit point. Let w e F be such a non-equilibrium limit point of the sequence {zn}. Then we use the following lemma proved in Smith (2005). Lemma. Let the sequence {(z,,, tn_i)} be derived using the algorithm above so Zn+i = z,, + unAn(zn), let w e intF, let w be a non-equilibrium so that V(w) > 0 and suppose that w is a limit point of {z,,}. Then, under our assumptions, there is a suffix i(w) such that V(ZJ(W)) < V(w). This lemma shows (since V(Zn) decreases) that w cannot in fact be a limit point of the sequence {z,,}. It follows that the sequence {zj has no non-equilibrium limit points. Hence, since F is closed and bounded, {zn} converges to the set of equilibria.
OPTIMISING PRICES Now we suppose that there is a specified smooth function Z which we here regard as a measure of total disbenefit. We seek to minimise Z at an equilibrium by charging prices for traversing certain arcs or routes in the network, thus influencing the equilibrium traffic distribution. If Z itself depends on the prices charged then charges will also influence Z directly. We suppose throughout that Z is a continuously differentiable and non-negative function of the route-flow vector X, the OD cost vector Y and the new route-price vector P. Adding a price vector P Suppose that route ijr is (possibly) to be subject to a charge Pjjr and that the vector P of all possible route prices P;jr will be confined to some polyhedral closed bounded set Fcontroi of feasible control vectors. In this case the vector (X, Y, P) will now be called a user-equilibrium if (X, Y) e R+N+K, P e Fcontroi, and for all ijr, Yjj - Cijr(X) - Pijr< 0 and Yy - C1]r(X) - Pijr < 0 implies Xijr = 0, and Dij(Y) - Ty(X) = 0. (5) To optimise, as before with equilibration, we need a bounded set of (X, Y) pairs. So our basic condition will now be enhanced to allow for the set of controls. We now assume that the following control-enhanced condition holds. CONTROL-ENHANCED BASIC CONDITION: There is a fixed set of N routes joining K OD pairs as above,
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C and -D are monotone and continuously differentiable, F = [0 N , UX]x[0 K , UY], and for all fixed P e F control , the set of equilibria is a non-empty subset of [0 N , !/ 2 UX]x[0 K , '/ 2 UY]. The previous V and A will now involve P. Now for all (X, Y, P) e FxF contro i: V(X, Y, P) = L ]r {Xi jr 2 [P, ]r + C ijr (X) - Y S] ] + 2 + (UX1]r- X ^ f Y s - (P1]r + C 1]r (X))] + 2 } + LjJY/tTijCX) - Dij(Y)]+2 + (UY1J-Y1J)2[Dij(Y) - Tij(X)] + 2 }, AV(X, Y , P) = (UX ljr - Xi,-r)2[Yij - CPijr + C i]r (X))] + - X ijr 2 [(P ijr + C ijr (X)) - Yjj]+, A2,j(X, Y, P) = (UYtJ -Ys)2[DgCY) - T i] (X)] + - Yg2[Tij(X) - D y (Y)] + , and
A(X, Y, P) = (A'(X, Y, P), A2(X, Y, P)).
The previous equilibrium constraint (X, Y) e F and V(X, Y) = 0 will now be written (allowing for the possible prices): (X, Y, P) e FxFCOTtrol and V(X, Y, P) = 0. We will also write: H = FxFcontroi- Finally we suppose that all the Nc functions h; are linear and let H = {(X, Y, P); hi(X, Y, P) < 0 for i = 1,2, 3 , . . . , Nc}.
A constraint qualification and new notation Suppose that (X, Y, P) e H is not an equilibrium so that V(X, Y, P) > 0. It follows that for this (X, Y, P), A(X, Y, P) is a feasible descent direction for V with P fixed and hence that: ProjF[-grad(x, Y) V(X, Y, P)] * 0 if V(X, Y, P) > 0. Here only (X, Y) varies in F and P is fixed in Fcontroi. It follows at once that: ProjH[-grad(x, Y, P) V(X, Y, P)] * 0 if V(X, Y, P) > 0. Here (X, Y, P) varies in H. Replacing (X, Y, P) by just x, the equilibrium condition (X, Y, P) e H = FxFcontrol and V(X, Y, P) = 0 becomes: x s H and V(x) = 0. Also (6) above may now be written: ProjH[-grad V(x)] * 0 if V(x) > 0.
(6)
(7)
This may be thought of as a constraint qualification applying (for any positive e) to the set EE={xeH;V(x)<s}. Condition (7) implies that if x is any point in the set Es such that V(x) = s and hj(x) = 0 for i = 1, 2, 3, . . . , k and no others then there is a direction 8 = ProJH[-grad V(x)] such that
Bilevel optimisation of prices in a variety of transportation models 9 S-descV(x) > 0 and 5-deschj(x) > 0 for i = 1, 2, 3, . . . , k. That is: there is a feasible descent direction for V on the edge of any Ee provided s > 0. Now the proposed optimisation algorithm, where (X1, Y1, P1) is any starting value for (X, Y, P)inHand(X 1 , Y1, P1) ^ (X2, Y2, P2) ^ (X3, Y3, P3) ^ , becomes: x1 is any starting 1 2 3 . value for x in H and x -^ x -> x A simple simultaneous descent direction (for x in the interior of H) We suppose that we wish to minimise Z(X, Y, P) = Z(x) subject to V(X, Y, P) = V(x) < s so that we need to optimise V and Z simultaneously in some way. In this paper we let x lie in the interior of H. We also need our control-enhanced basic condition above. We make the further initial basic assumption that gradZ(x) ^ 0 for all x e H. Now, under our control-enhanced basic conditions, V(x) > 0 implies gradV(x) ^ 0. So we may now let (for x e H and V(x) > 0): descV(x) = - grad V(x)/||grad V(x)||, descZ(x) = - gradZ(x)/||gradZ(x)|| and desc(Z, V)(x) = /2descZ(x) +'/2descV(x). (8) This direction (8), if non-zero, reduces V and Z simultaneously. In any case this direction is never an ascent direction, for either V or Z. However descV is not defined where V = 0 and changes sharply in the vicinity of a point at which V = 0. So it is natural to change this direction (8) slightly so that it is defined everywhere and is smoothly varying. This may be done by enlarging the equilibrium set E. We are led to put (where s > 0): Es = {x e H: 0 < V(x) < s} and for all x e H: A£(x) = [ 1 -V(x)/e]+descZ(x) + [V(x)/s]desc(Z, V)(x) + [V(x)/e-1 ]+descV(x). (9) Our general approach here is also similar that adopted in Clegg and Smith (2001). The present direction (9) with the present definition of the simultaneous descent direction desc(Z, V)(x) has been introduced so as to allow an effective step length selection procedure to be more easily designed. [Estimation of Ae(x) is made more simple by supposing that the cost and demand functions have symmetrical Jacobians.] For any x in the interior of H the zeros of As(x) coincide exactly with points x in EE at which Z is stationary; or those points in E£ for which there is no descent direction for Z which remains inside Ee. We will now show this. Optimality conditions for x in the interior of H
Definitions ofs-feasible descent and s-linear optimality (for x in the interior ofH) The vector u is an s-feasible Z-descent direction at x (in the interior of H) if and only if
10 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction u-[descZ(x)] > 0 and either V(x) < s or {V(x) = s and u-[descV(x)] > 0}. x* (in the interior of H) is s-linearly-optimal if and only if V(x*) < s and there is no s-feasible Z-descent direction at x*. s-linear-optimality conditions (in the interior ofH) In this section we give optimality conditions for any x in the interior of H. These results are particularly useful if H has been chosen correctly; so that at any e-linearly-optimal x = (X, Y, P) the upper and lower bounds on x = (X, Y, P) are not binding. For such x, in the interior of H, we let A6(x) be given by (9) above and show how this vector may be used to classify points x according to whether they are s-linearly-optimal or not. Observe that desc(Z, V)(x) = !/->descV(x) + V2descZ(x) (if non-zero) is a direction in which both V and Z decline; and is never a direction of increase for either V or Z. So if xeintH and !4descV(x) + V2descZ(x) is non-zero then x is not E-linearly-optimal and also Ae(x) reduces both V and Z. This is the crux. It follows from these conditions that, at least for x s intH, AE(x) = 0 if and only if x is slinearly-optimal. In detail, there are four mutually exclusive cases (still supposing that x is in the interior of H): 1. V(x)>s, 2. 0 < V(x) < s, 3. V(x) = s and desc(Z, V)(x) * 0, and 4. V(x) = s and desc(Z, V)(x) = 0. We show that x is not E-linearly-optimal in cases 1 - 3 and A s (x) ^ 0 then; and that, in case 4, x is E-linearly-optimal and also that in this case A£(x) = 0.
1. V(x)>s In this case x ? E E and also Ae(x) = [V(x)/s]desc(Z, V)(x) + [V(x)/s - l]descV(x) is non-zero as descV(x) is non-zero and desc(Z, V)(x) is never a direction in which V ascends. Of course x is not s-feasible and so is not E-linearly-optimal. Here following As(x) reduces V. 2. 0 < V(x) < s In this case x e Ee and Ae(x) = [V(x)/s]desc(Z, V)(x) + [1 - V(x)/s]descZ(x) is again non-zero as descZ(x) is non-zero and desc(Z, V)(x) is never a direction in which Z ascends. Here following As(x) reduces Z maintaining V < s. 3. V(x) = s and desc(Z, V)(x) * 0 In this case x e Ee and Ae(x) = desc(Z, V)(x) is non-zero and so is a simultaneous descent direction for both V and Z. Thus As(x) is an Efeasible Z-descent direction at x so x is not s-linearly-optimal. Here again following As(x) reduces Z maintaining V < s.
Bilevel optimisation of prices in a variety of transportation models 11 4, V(x) = s and desc(Z, V)(x) = 0 In this case also x e Es. Now desc(Z, V)(x) = 0 and so descV(x) = - descZ(x). Consider any Z-descent direction u. Then u-[descZ(x)] > 0 and so u-[descV(x)] = u-[- descZ(x)] < 0, and u is not an s-feasible direction at x. Thus there is no s-feasible Z-descent direction from x and also V(x) < s, so x is s-linearly-optimal. In this case As(x) = 0. Thus we have shown that (at least for x e intH) zeros of A8(x) coincide with points x e Es at which there is no s-feasible descent direction for Z at x. Such points are e-linearly-optimal. We have also shown that A6(x) is an H-feasible descent direction for V if x is not in Es and that Ae(x) is an Es-feasible descent direction for Z if x is in EE and is not s-linearly-optimal. Thus, at least for x in the interior of H, As(x) is an excellent arbiter of s-linear-optimality at x; and for those x which are not s-linearly-optimal indicates a sound direction for moving x. An (xn, tn.i)-updating dynamic Armijo-like optimisation algorithm Here we change the previous equilibration algorithm to create an optimisation version. Now we suppose that if we are at iteration n; at a non-s-linearly-optimal xn e H where the search direction is Ae(xn) and the step length actually used at xn is un then our next x will be xn+i = Proj(xn + unA(xn)) where Proj now denotes projection onto H. Initially we suppose that if un is any used step length then Proj(xn + unAe(xn)) = xn + unAe(xn). That is, we suppose that the projection operator does not actually do anything; and so the boundary of the feasible set H is here having no effect. Let Gi(x) = - gradV(x)-Ae(x) and G2(x) = - gradZ(x)-Ae(x). These are continuous functions of x defined for all x e H. Both must be estimated using a two-point estimation where the second point lies in the direction A6 from the first. For our purposes here we specify a simple dynamic Armijo-like optimisation scheme based on the previous dynamic Armijo-like equilibration scheme but now taking account of the two objectives V and Z and the continuous functions Gi and G2 above, instead of just the previous V and the previous G. As before we update (xn, tn.i). To be specific, we start at an arbitrary xi e H and to = 1. Then if we are at a current non-s-linearly-optimal point xn and the previous possible step length was tn-i then we are to update xn and tn_i (for n > 1) as follows. To be most general we will, in this algorithm statement, let V[y] = V(ProjH(y)) for all y e RN + K. Let V(Xn) > e. In this case we follow the equilibration algorithm above with G replaced by Gi.
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Let V(x^ < e. Firstly, x n is kept fixed and V i is halved to obtain the sequence tn_i, (54)tn.i, ('/2) 2 t n .i,...., (Vytn.i where the halving ceases as soon as: Z[x n + (MOV.AeCx,,)] - Z(x n ) < - (l/8)C/ 2 ) p t n .iG 2 (x n ) and V[x n + (>/2)ViAE(xn)] < s for the first time, p = 0 is allowed here as before. The halving surely ceases by definition of G2 and the Z-descent property of Ae maintaining V < s, because xn is not s-linearly-optimal. Then let un = (Viftn-i for this p (this is to be the used step length at x n ) and xn+i = x n + u n A s (x n ). Finally update tn_i as follows: if Z[x n + unAE(xn)] - Z(x n ) < -3/4UnG2(xn) put tn = 2u n ; if-3/4UnG2(xn) < Z[x n + unAs(xn)] - Z(x n ) < -1AunG2(xn), put tn = un; and if-'/4UnG2(xn) < Z[x n + unAE(xn)] - Z(x n ) < - (l/8)u n G 2 (x n ) put tn = Vox*. These are three mutually exclusive possibilities and they together exhaust all eventualities since, by choice of u n = (Vi)ptr,--\, Z[x n + unAE(xn)] - Z(x n ) > - (l/8)u n G 2 (x n ) is not possible. The algorithm is terminated as soon as V(x n ) - s and G 2 (x n ) are both less than preassigned tolerances.
CONVERGENCE TO A STATIONARY POINT We assume that our control-enhanced basic condition holds.
Convergence preliminaries For any starting point xi e H and with an initial step length to = 1 we suppose that the algorithm specified above generates an infinite trajectory: xi, x 2 , X 3 , . . . , xn> . . . . and an infinite sequence to, ti, t2, t 3 , . . . , tn, . . . . of possible step lengths. To prove convergence in this case we will use proof by contradiction. So we suppose that {xn} is a given infinite sequence generated by our algorithm which does not converge to the set O e of slinearly-optimal points. Then not all limit points of the sequence {xn} belong to OE and since H is closed and bounded there is a limit point w which is not in OE. Thus we now make the basic assumption that w is not s-linearly optimal and is also a limit point of {x n }. We will show that this leads to a contradiction. This will show that, in fact, if the sequence {xn} is generated by the algorithm then all limit points of the sequence {xn} are in OE and so (as H is closed and bounded) dist(xn, O6) -> 0 as n -> 00.
Theorem Let our control-enhanced basic condition hold. Let {xn} be the sequence above. Suppose that {xn} lies in the interior of H, let x* be the limit of any subsequence of the above sequence {xn}, and let x* e intH. Then x* is s-linearly-optimal. [Note: these interior conditions may well happen naturally, especially within stochastic assignment models.]
Bilevel optimisation of prices in a variety of transportation models 13 Proof Suppose that x* is the limit of a subsequence of the above sequence and that x* e intH. We will show that none of the following three alternatives in 7.4.2 can occur: (l)V(x*)>s, (2) 0 < V(x*) < s, (3) V(x*) = s and desc(Z, V)(x*) * 0. It will then follow that (4) above holds, and so x* is s-linearly optimal. Ruling out (I) This is essentially as before with equilibration. Ruling out (2) Suppose that (2) does hold or that V(x*) < s. Then Ae(x*) = [V(x*)/s]desc(Z, V)(x*) + [l-V(x*)/s]+descZ(x*) + [V(x*)/e-l]+descV(x*) = [V(x*)/s]desc(Z, V)(x*) + [l-V(x*)/s]descZ(x*) * 0. [l-V(x*)/s]descZ(x*) is a descent direction for Z at x* (and desc(Z, V)(x*) is never ascent) so [gradZ(x*)]-AE(x*) < 0. A small enhancement of the equilibration argument now works in this case too, but with Z instead of V, ruling out (2). Ruling out (3) Suppose that (3) does hold or that V(x*) = s and desc(Z, V)(x*) is non-zero. Then A6(x*) = [V(x*)/s]desc(Z, V)(x*) + [l-V(x*)/s]+descZ(x*) + [V(x*)/e-l]+descV(x*) = desc(Z, V)(x*) * 0, and this (being non-zero) is a descent direction for both V and Z at x*. An enhancement of the equilibration argument embodied in the lemma in section 6 above now works in this case too, but this argument must in this case be applied to both Z and V. See Smith (2005) for the details. This means that x* cannot be a limit point and rules out (3). It follows from the arguments above that if x* is any limit point in the interior of H, (4) above must hold or: V(x*) = s and desc(Z, V)(x*) = 0. So x* is e-linearly-optimal, as we have seen from the s-linearly-optimality conditions above in section 7.4.2. and the proof is complete. Allowing for the boundary of H The boundary of H may be taken account of by re-designing the search direction Ae. We change descV(x), descZ(x) and desc(Z, V)(x) for those x close to the boundary of H. The new versions will be called descHV(x), descHZ(x) and descH(Z, V)(x) and Ae will become AHe. We do not give these details here. They are given in Smith (2005).
14 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
APPLICATIONS TO DIFFERENT MODELS The basic structure exploited above is as follows: there is a function O, defined on the product set FxFcontroi. such that - 0 for all ijr. Xjjo is essentially a slack variable added in to model 0. The rigid demand problem represents the given variable demand problem and is driven by the original costs Cyr(X) and also the new opportunity cost Cij0(X) = D^EeiX^) = D"!(T - Xij0). It is easy to check that the whole new cost function C is monotone if the original C and the original -D are both monotone. Thus we may add a price vector as previously and utilise the same simultaneous descent algorithm to optimise this equilibrium model. Equilibrium model 2: extra route when the inverse demand function is unknown The set of "flows" in this model is the set of all (Xo, X, Y) such that (Xo, X) e {(Xo, X); Er Xijr = Ty for all ij and Xijr > 0 for all ijr} where in the sum here r runs over 0, 1, 2, 3 , . . . Ny, and Y s {Y; 0 < Yy < Uy}. A typical "route-flow" vector is now (Xo, X, Y) where: (Xo, X, Y) e {(Xo, X); Zr Xyr = Ty and Xijr > 0 for all ijr}x {Y; 0 < Yy < Uy}. With this set of feasible "flows" this "rigid demand" model is driven by "costs" as follows: Yy: this cost is felt by each element of the flow Xyo, Cyr(X): this cost is felt by each element of the flow Xyr for r = 1, 2, 3 , . . . , Ny, and Ty - Xyo - Dy(Y): this "cost" is felt by Yy. Assuming for simplicity that all the listed routes are used at equilibrium: Yy and all Cyr(X) are equal so that the flows Xyi,. . ., Xyuy have no incentive to change and also Y represents these OD costs; all Ty - Xyo - Dy(Y) = 0 and so Yy e [0, Uy] also has no incentive to change.
Bilevel optimisation of prices in a variety of transportation models
15
Thus at an interior equilibrium as specified above the Wardrop condition holds, and, since Y does then represent the OD costs experienced on all routes, Ty - Xij0 - Djj(Y) = 0 implies that the total ij flow Tij - Xjjo on the real network does equal Djj(Y) as it should. Monotonicity of C and -D now implies that: (Y, C(X), T - Xo - D(Y)) is a monotone function of (Xo, X, Y). This means that the optimisation method designed for model 0 also works here. This formulation may of course be used to restate, in a more complicated way, a given rigid demand problem; just "reformulate" the given rigid demand problem as a variable demand problem with a flat (zero slope) demand function by making Dy(Y) = a constant Djj for all Y. Equilibrium model 3: A responsive control policy in the Payne-Thompson model In order to study ramp control on freeways, Payne and Thompson (1975) introduced an equilibrium model with queueing delays at bottlenecks. In their model as long as a bottleneck is saturated the queueing delay at the bottleneck is independent of flow, and is represented by an independent non-negative variable determined by the equilibrium conditions; but if the bottleneck is unsaturated then the bottleneck delay must be zero. Bringing in the delays at bottlenecks in this way, as separate variables, allows ramp metering to be modelled sensibly. The "new" independent bottleneck delays are here added to costs arising from more standard cost-flow functions ca which may be thought of as applying to the rest of arc a. We here consider the Payne-Thompson model with capacity constraints and explicit queueing delays. We insert signal green-times as in Smith (1987). There can be great advantages in real traffic situations if the signal controls respond appropriately and automatically to the flows and delays on the network and traffic responsive signal control is so often utilised in practice. This extends the type of equilibrium model which can be considered. Certain subsets of arcs a comprise stages; any stage contains only arcs a terminating at a single node and all arcs in a given stage are given green if the stage is green. The kth stage at node i will be called stage ik. Let va sa ba Qik ca(-)
be the traffic flow along arc a, be the saturation flow at the exit of arc a (both in vehicles per minute), be the bottleneck delay at the exit of arc a (in minutes per vehicle), be the proportion of time stage ik is green and be the cost function for arc a (this excludes the bottleneck delay at the arc exit).
Then we make the following agreements: the flow along arc a — V a ( X ) — 2-those ijr passing through arc a X j j r ;
the non-bottleneck cost of travel along route ijr ~~ ^ i j r ( X ) — 2^arcs a which belong to route r joining OD pair ijC a (V a );
the bottleneck delay on route ijr
16 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
)
Z_iarcs a which belong to route numbered r joining OD pair ijD a ,
the "pressure" felt on stage ik by the responsive signal setting algorithm =
Jik(b) — Z^arcs a which belong to stage ik Sa^ai a n a
the green-time awarded to arc a Qa
^--stages ik which include arc a Wik-
The special responsive policy PO seeks to equalise the pressures felt on the various stages by suitable allocations of green-time. Thus if one stage has a higher value of J than another then PO will reallocate some green-time from the less pressurised stage to the more pressurised stage. At equilibrium all stages with a positive green time have equal values of J and stages with smaller values of J have no green-time. This equilibrium allocation of green-time is similar to the Wardrop allocation of traffic flow to routes and is given by the fourth and fifth condition in the equilibrium condition below. The grand equilibrium flow/cost/queue/green-time equilibrium condition comprises the following conditions. These must hold (for a fixed price vector P): for all nodes i, j , routes ijr, arcs a, and signal stages ik. Yy - Cijr(X) - Bljr(b) - Pijr Dy(Y) - Ty(X) va(X) - saqa Jik(b) - m;
< = <
y 2 ) may be preferred.
30 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In the case that the approximate O-D demand functions are tuned by updating a set of parameters 0 e 0 directly, the problem is formulated as the following similar BLPP: mm F0D(Q) = Fi(v(Q),v) + F2(d{Q),d)
(15)
where (v(Q),d(6)) solves the following elastic-demand UE problem for given 0 6 0 and
xef: va
r™PQZj('»
+T
va
dw
« ) d ( 0 + £ J)dcn-£J|;;1(a,e)d(»
(16)
The same quadratic function in (14) can be used for the objective function (15). The above iterative bilevel O-D demand function estimation and bilevel link toll optimization process is repeated until an appropriate convergence criterion is reached. To sum up, the general iterative trial-and-error procedure through sequential price experimentation for estimating the second-best link tolls is given below. Trial-and-Error Second-Best Toll Adjustment Scheme: Step 0.
(Initialization) Let \gw [cw,Q I, w e W\ be an initial set of approximate demand functions. Set k = 0 .
Step 1.
{Estimate link tolls) Solve the BLPP (8)-(9) to obtain a new set of link tolls
{ 50 mph) = 0.79, P(v < 50 mph) = 0.21.
108 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
Figure 2 (a) Distributions of VMT and VHT vs. speed for I-10E and (b) A two-regime BPR model.
IDEAL METERING Figure l(b) suggests that holding back volume surges by metering on-ramps may prevent the occurrence of the congestion regime at some bottlenecks, and figure 2(a) implies a large reduction in delay if this can be done. Designing a ramp metering algorithm for a specific freeway section is arduous. Many set points and feedback gains must be selected (Papageorgiou, M., 1983; Papageorgiou, M. et al., 1991), based on a calibrated simulation model. But there is a simple procedure to roughly estimate the benefits from ramp metering without detailed simulations, based on the hypothesis that the congestion regime can be avoided by controlling flow according to the Ideal Metering Principle (IMP) (Jia, Z. et al., 2000): If volume surges at on-ramps are held back by a metering policy that always keeps flow below its capacity in every link, freeway speed will be maintained at 60 mph and congestion will not appear. As a consequence of metering vehicles may be stopped at the ramps for some time. The IMP hypothesis has two parts. One part is that if flow is always maintained below capacity, or equivalently, if density is always less than critical (pcr), traffic will be kept in the free flow regime. Data like in figure l(b) provide indirect support: If the traffic density is never allowed to enter the critical region, traffic will always stay in the free flow regime. The definition of 'capacity' is empirical: It is taken to be (say) 95% of the maximum sustained observed flow. The second part of the hypothesis is that maximum flow occurs at free flow speeds, nominally 60 mph, as in Los Angeles (Jia, Z. et al., 2001) and Orange County (Chen, C. and P. Varaiya, 2001). Of course not all congestion is due to volume surges at on-ramps and practical considerations, such as ramps of insufficient length, may prevent implementation of a proper metering policy. The planner should ask: "What will be the impact of implementing IMP-conforming ramp metering if the IMP hypothesis is true?"
Empirical assessment of traffic operations 109 A procedure to answer this question is illustrated in (Jia, Z. et al., 2000), using data for a 7mile section (postmiles 0-7) of I-405N in Orange County, during 0500-1000 for 10 weekdays in June 1998. The section is divided into 13 links, each corresponding to one VDS; eight links have one on- and off-ramp each. A virtual on-ramp is created at the beginning of the most upstream link in order to account for metering of on-ramps upstream of the study section.
Figure 3 The top graph is the time in VHT actually spent on the freeway section, every 5 minutes. Units are normalized to VHT per hour, so the total VHT on this section, between 0500 and 1000 is the area under the top graph. The middle graph is the VHT per hour under IMP metering, including time in ramp queues. The bottom graph excludes time spent on the ramps, so it is the VHT per hour that would be spent traveling at 60 mph. The area between the top and middle graphs is the time saved by metering. The area between the middle and bottom graphs is the time spent at the ramps. Source: (Jia, Z. et al., 2000). The capacity of each link is calculated as the maximum sustainable aggregate flow. Inflows at on- ramps and exit flows at off-ramps are assumed to remain unchanged despite the metering, whose impact is estimated as follows: 1. At each on-ramp, inflow is metered so that the link flow remains 5% below the link capacity; 2. Traffic on each link after metering is assumed to move at 60 mph; 3. The queue at each on-ramp is calculated by accumulating the net inflow. On average, two-thirds of the total delay (defined as additional vehicle-hours traveled (VHT) driving below 60 mph) is eliminated by ramp metering. More insight is gained from figure 3, which shows how metering holds back large surges in demand.
110
Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
This procedure was repeated in (Chen, C. et al., 2001) for five freeways (1-5,1-10, US 101,I110 and 1-405) in Los Angeles during 0000-1200, 3-9 October, 2000. That exercise found that IMP-metering reduces delay by 70%. PeMS calculates the total congestion delay (from driving below 60 mph) for Los Angeles for 2003 to be 83 million vehicle-hours. The procedure suggests that ramp-metering may eliminate 57 million vehicle-hours of delay, which at $20/veh-hr is in excess of 1 billion dollars. Even if only one-half of the delay savings from IMP-metering can be practically realized, this indicates the large productivity gain that good management can achieve.
BOTTLENECKS Bottlenecks can cause congestion, which may be reduced by ramp metering. A bottleneck may be associated with physical features such as ramps, lane drops, grade changes, curvature, lane closures, and accidents; but traffic jams and congestion may 'spontaneously' arise in locations with none of these features. In the absence of a guide to locating bottlenecks and estimating their severity, we need an algorithm to automatically (1) identify all bottlenecks, and (2) calculate the delay each one causes. Such an algorithm is reported in (Chen, C. et al., 2004b), and applied using flow and speed data from 263 VDSs on 270 miles of seven freeways in San Diego. The algorithm uses a sustained speed gradient between a pair of upstream-downstream detectors to identify bottlenecks. We describe the algorithm. Consider a freeway with n detectors indexed i = 1,..., n, each giving speed and flow measurements, averaged over 5-minute intervals indexed t= \, 2,... Detector i is located at postmile x,-; v,-(f) = v(x,-, t) is its speed (miles per hour, mph) and qi{t) = q(xt, t) is its flow (vehicles per hour, vph) at time t. If*,- < Xj, it is understood that x, is upstream of x,. The algorithm has four steps. First, it declares an active bottleneck at certain locations and times if the data meet criteria (l)-(4) below. Second, it includes additional time periods as part of the same bottleneck activation, provided nearby time intervals are selected in the first step. The criterion for this is (5). Third, it calculates the delay caused by a bottleneck, using (9). Lastly, identified bottlenecks are ranked in terms of frequency of occurrence and severity to isolate recurrent from transitory bottlenecks and to help prioritize mitigation efforts. Step 1 Declare an active bottleneck between locations x,- < xj during t if all four inequalities hold: Xj -
2 miles,
v(xk, t) - v(xi, t)>0 ifxi <xk<xi<xh
(1)
(2)
V{XJ, t) - v(x.i, t) > 20 mph,
(3)
v(xh t) < 40 mph.
(4)
Thresholds in (l)-(4) are selected on the basis of experience. In Los Angeles free flow speed is 60 mph and, when a bottleneck is activated, speed drops rapidly to below 40 mph (e.g. figure l(b)). Hence the 20 mph minimum speed differential (3) and 40 mph congestion speed (4) thresholds. The maximum separation of 2 miles in (1) is designed to include locations where speed continues to drop as we go downstream, but the difference between each neighboring pair is small. Location xt is upstream of Xj, but there may be other detectors at xt,
Empirical assessment of traffic operations 111 x; between these locations. The constraint (2) that speed should drop continuously is the algorithm's characterization of an active bottleneck. Step 2 Sustained bottlenecks last longer than five minutes. Let A,{t) = 1 if there is an active bottleneck at location i and time period t; otherwise At(i) = 0. A bottleneck is sustained between times t\ and ?2 if ^4-(r)>9Ar,V?, 0
(6)
where a
= shape parameter
[-]
132 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction (3
= scale parameter
[-]
The two examples in Fig. 3 show that the Weibull distribution fits very well into the PLM estimation. Of course, the Product Limit Method for capacity estimation could also be used with traffic densities k instead of volumes q. Eq. 4 and 5 remain unchanged, except that q is replaced by k. It was expected that k would be a better determinant for the occurrence of a breakdown than q and that, consequently, the analysis would reduce the variability of the resulting distribution function. This was tested by Regler (2004) using data from six 3-lane freeway sections. The median of the breakdown densities ranged from 70 to 90 veh/km with Weibullparameters a = 8.4 through 13.2 and p = 72 through 92 veh/km for the analysis of 5-minute intervals. The resulting distribution of breakdown density and breakdown capacity tended towards larger variances compared to the analysis over the q-axis. Therefore, it was not advantageous in this context to estimate capacity distributions from densities. Moreover, densities are artificial parameters that must be calculated from measured speeds v and volumes q (k = q / v). The k-based analysis also needs a more complicated definition of the speed threshold between fluent and congested traffic. Therefore, using the Product Limit estimation based on densities is not recommended.
APPLICATION TO FREEWAYS So far, only results from freeway bottlenecks according to Fig. la have been discussed. Under this assumption, the estimation technique would be restricted to rather specific geometric situations. Regler (2004) applied the methods described above to freeway sections without a distinct bottleneck (Fig. lb) whose geometric properties included no change in the number of lanes. To make sure that intervals with a tailback from downstream bottlenecks did not impair the results, classification C2 (see above) was considered as well. With this technique, the capacity distribution of quite a variety of freeway sections could be analyzed (Table 2). The analysis was based on 5-minute counts taken over several months, with an arrangement as shown in Fig. lb. All sites are on 3-lane freeway carriageways, mainly in level terrain. Periods of work zones were excluded from the data. It turns out that the shape parameter a in the Weibull distribution typically ranges from 9 to 15 with an average of 13. This magnitude seems to be characteristical for 3-lane freeways. The scale parameter p of the Weibull distribution varies over a wide range between the analyzed sections. This may be mostly due to different geometric and control conditions, different driver and vehicle populations, and diverse prevailing travel purposes (long distance travel versus metropolitan commuter traffic).
Reliability of freeway traffic flow 133 Table 2: Parameters p and a, expectation E(c) and standard deviation c of the estimated Weibull capacity distribution at 15 freeway sections (3-lanes, 5-minute intervals) Section A3-1 A5-1 A5-2 A5-4 A5-5 A5-6 A5-7 A5-8 A9-1 A9-2 A9-3 A9-4 A9-5 A9-6 A9-7
P [veh/h] 7441 6217 6074 6608 6392 6272 7194 6884 7937 7399 5988 6141 6648 7109 6648
a[-] 11,31 11,15 13,59 13,92 14,16 14,69 13,98 13,35 8,85 13,66 14,82 18,86 14,24 9,62 14,92
E(c) [veh/h] 7115 5941 5847 6365 6161 6053 6932 6622 7510 7124 5780 5969 6409 6752 6419
a [veh/h] 762 645 526 559 532 505 606 606 1013 637 478 392 551 842 528
Having found that the shape parameter a seems to be almost constant, we may transform the capacity distribution function to fit different interval durations A. According to Eq. 1, F5(q) is the probability of a breakdown during A = 5 minutes at flow rate q. Hence, (1 - Fs(q)) is the probability of no breakdown occurring in this interval. If we assume that breakdowns occurring in succeeding intervals are independent of each other, then the probability of fluent traffic flow during a whole hour is p60 (fluent traffic) = [l - F5 (q)] n
(7)
Using the Weibull distribution (Eq. 6), this is converted into i2
F « ( q ) = l - P 6 o (fluent traffic) = l - e
(-T
UiJ
(8)
which is again a Weibull distribution with an unchanged shape parameter a and a scale parameter p6o = r p 5 , where r = 12("1/a). With a = 13 (see above) we get r « 0.82 » 1/1.2. This means that for 5-minute observations the expected capacity should be in a range of 1.2 times the 1-hour capacity. This factor of 1.2 seems to be typical for the transformation of capacities from 5-minute intervals into 60-minute intervals, as was pointed out by Keller and Sachse (1992) or Ponzlet (1996) based on empirical capacity estimates obtained from the fundamental diagram. One might object to Eq. 8 that the traffic volume q usually is not constant during a whole hour. However, numerical calculations showed that volume variations during one hour did not
134 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction significantly change the results. It also seems to be realistic to assume that traffic breakdowns in succeeding intervals are independent of each other since there is no imaginable reason why the opposite should be true. This question may, however, be made a subject of further research. The new technique was used to investigate differences in performance between dry and wet road surfaces. At all sections under examination (see list in Table 2), it turned out very clearly that on a wet road surface the capacity was reduced by around 11 %. The effects of darkness were investigated as well. Contrary to Ponzlet's (1996) results, it was clearly found that darkness did not shift the capacity distributions. The results shown in Table 2 also demonstrate differences in the capacity distribution between an uncontrolled freeway (section A9-3) and a freeway with traffic adaptive variable speed limits (section A9-4): The mean capacity of the controlled section is slightly (by 3 %) higher compared to the uncontrolled section, but the standard deviation is significantly lower (cf. Fig. 4). The two analyzed sections are the two opposite carriageways of the freeway A9 near Munich and thus have similar geometric and traffic characteristics.
Fig. 4: Capacity distributions for a 3-lane freeway with and without variable speed control (13.5 % average truck percentage, 5-minute intervals) All these examples demonstrate that the statistical interpretation of freeway capacity together with the corresponding estimation technique provides a better understanding of freeway traffic operation. It improves the methodology of investigating differences between various external conditions.
FREEWAY TRAFFIC DYNAMICS So far, only capacities in the upper branch of the speed-flow diagram, i.e. under fluent traffic conditions, have been analyzed. However, the lower branch of the speed-flow diagram representing congested conditions must be considered as well.
Reliability of freeway traffic flow
135
It is well known that dynamics in the speed-flow diagram (i.e. the sequence of v-q points over time) follow specific patterns. The first to report typical hysteresis phenomena within these dynamics were Treiterer and Myers (1974). More recently, Kim and Keller (2001) came to the conclusion that six different typical traffic states should be distinguished within the speedflow diagram (or the fundamental diagram). These dynamics were analyzed by Regler (2004) based on 5-minute data for the freeway sections A3-1 and A5-7 (cf. Table 2) extending over 4 months and 10 months, respectively. More than 120 breakdowns from fluent traffic to congested traffic were observed. The analysis came to the conclusion that there are three different states of traffic conditions to be distinguished: 1. Fluent traffic at high speeds (i.e. v > 70 km/h) and low densities. In this state, volumes q may range from 0 to the maximum flow rate. This is the ascending branch of the q-k diagram. 2. A transient state with an average velocity of around 60 km/h and rather high volumes. We like to call this state "synchronized flow", knowing that this term is used by other authors (Kerner and Rehborn, 1996) with a slightly different meaning. In this state, vehicles are forced to travel at fairly similar speeds on all lanes. 3. Congested traffic with low speeds and low traffic volumes. To illustrate typical dynamic patterns, Fig. 5a shows observations from the freeway A5 with only states 1 and 2 involved. In all examples, the transition from fluent to synchronized flow began at a rather high volume. Traffic flow stabilized at slightly lower volumes at an average speed of about 60 km/h. From here, the traffic flow recovered to fluent traffic conditions. All recoveries involved much lower traffic volumes than the preceding breakdown. This hysteresis phenomenon seems to be a characteristic of traffic dynamics. In addition to these 2state sequences, Fig. 5b shows those cases from the A5 where a breakdown from fluent to synchronized flow was followed by a subsequent transition into congested traffic with very low speeds. The recovery back to fluent traffic did never happened directly. Instead, each recovery process passed through the transient state of synchronized flow.
136 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
Fig. 5: Two typical patterns of traffic dynamics during breakdown and recovery (freeway section A5-7, 5-minute flow rates) Empirical analysis of freeway traffic dynamics showed that: Transitions between traffic states usually happen suddenly, i.e. within rather short times and distances. Breakdowns from fluent traffic are first followed by the synchronized traffic state. From there, speed and flow rate may decline further to heavy congestion. All recoveries pass through the synchronized state. No recovery process jumps directly from congestion back to fluent traffic. Recovery from synchronized to fluent traffic always involves much lower volumes than the breakdown. The difference between breakdown volume and recovery volume (fluent traffic after a recovery) on the observed 3-lane freeway sections ranged from 500 to 1500 veh/h. Volumes in the synchronized state are always lower than the maximum flows in fluent traffic (= capacity). Observed differences on 3-lane freeways ranged from 200 to 600 veh/h, measured in 5-minute-intervals. This effect is called "capacity drop" (cf. following section). Because of the large amount of data that was analyzed and because of the remarkable analogy between all cases observed, we may say that these properties are typical for the dynamics of freeway traffic flow. It should, however, be admitted that these dynamic effects are predominantly due to driver behavior, concerning headways at high speeds in dense fluent traffic, together with braking and accelerating behavior. Therefore, different typical dynamics might be found for other driving cultures than in Germany. Even here, under variable speed control, a few traffic breakdowns were observed where the average flow in synchronized traffic was higher than the flow rate before breakdown.
CAPACITY DROP A number of investigations has proven the existence of different capacities under flowing and congested traffic conditions. Banks (1990) as well as Hall and Agyemang-Duah (1991) analyzed this "capacity drop" phenomenon for different North-American freeways. Capacity
Reliability of freeway traffic flow 137 drop values of between 3 and 6 % were measured. Ponzlet (1996) analyzed traffic flow on German freeways to see whether this phenomenon existed. He determined a 6 % drop for 5minute flow rates. Brilon and Zurlinden (2003) analyzed the capacity drop by comparing the stochastic capacity to flow rates in congested flow. They computed an average of 24 %, which is very high compared to other authors' results. There are different hypotheses about the reasons for the capacity drop phenomenon: Bottleneck downstream of the study site: The flow at the point under investigation will remain fluent until the section between this point and the bottleneck is filled with congested flow. After this time, the maximum flow will be the bottleneck's capacity. Different driver behavior: Drivers in fluent traffic accept shorter headways since they expect to be able to pass the vehicles in front. Once they have given up this idea, they switch to a more safety-conscious style of driving and keep longer headways. Restricted acceleration capabilities: At the front of the congested area, drivers need to accelerate. Some vehicles, however, have limited acceleration power, which opens a larger gap in front of them. These hypotheses and the value of the capacity drop were analyzed by Regler (2004) for the 15 freeway sections listed in Table 2, using different approaches. The main question was: What are the capacities under synchronized flow conditions as described in section 4? The conventional traffic flow model of van Aerde (1995) as enhanced by Ponzlet (1996) to account for the capacity drop phenomenon yielded an average drop of 270 veh/h in 5-minute flow rates. Using a distribution of breakdown flow rates (observed immediately prior to breakdown) and a distribution of queue discharge flow, an average drop of 250 veh/h in 5minute flow rates was determined, which is comparable to the result from the fundamental diagram. It is not easy to find a distribution of capacity after a breakdown that is adequate to the Product Limit estimate (Eq. 4). Brilon and Zurlinden (2003) applied a distribution of all flow rates in congested traffic and computed a very high capacity drop. Regler (2004) developed a method comparable to the Product Limit technique to obtain a capacity distribution in queue discharge flow. This method is based on the following assumptions (cf. Fig. 6): Flow rates during congestion do not represent the maximum possible flow of congested traffic. Capacity can directly be measured in queue discharge flow, i.e. in the last 5-minute interval before recovery of traffic flow (uncensored data). Flow rates during congestion are lower than capacity in queue discharge flow (censored data). Flow rates in free flow are not relevant for the capacity of queue discharge flow.
138 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
Fig. 6: Flow rate and speed time series during congestion (freeway A5, 5-minute intervals) Based on these considerations, the observed flow rates in each time interval can be classified as follows: B*: Traffic recovers from congestion to free flow, i.e. the average speed exceeds the threshold value from time interval i to interval i + 1 F*: Traffic is congested in intervals i and i + 1, i.e. the average speed is lower than the threshold value in both intervals. This interval i contains a censored value. C*: Traffic is fluent in interval i, i.e. the average speed is above the threshold value. This interval is not relevant. After this classification, a capacity distribution for queue discharge flow according to Eq. 4 (where {B} is to be replaced by {B*}) can be computed from empirical data. This capacity level turned out to be always lower than the capacity before breakdown obtained from Eq. 4. The difference between both distributions (see Fig. 7), represented by the median value, for instance, may be regarded as the capacity drop.
Fig. 7: Capacity distributions for pre-queue (cf. section 2) and queue discharge flow (freeway section A5-7, 11.8 % average truck percentage, 5-minute intervals)
Reliability of freeway traffic flow
139
For the 15 freeway sections listed in Table 2, an average drop of 1,180 veh/h was estimated using this algorithm. It should, however, be mentioned that the results for the capacity drop varied widely between the sites investigated. All attempts to identify regularities within the variation failed. It might be that the capacity drop shows some chaotic properties, as was indicated by other authors (Kerner, 2000) using different methodologies.
TRAFFIC RELIABILITY Traffic reliability is an important factor for the assessment of the performance of highway segments and systems. In this context, the term "reliability" mainly refers to the variability of travel times. However, several definitions can be found in the literature. A comprehensive outline of these definitions is given by Shaw (2003). Here, traffic reliability is assessed by analyzing the probability that a freeway link is not congested, i.e. that the travel time does not exceed an acceptable level. This is becoming a question of increasing importance for "just-in-time" transportation in modern logistics chains. With the stochastic concept of capacity, it is possible to assess this kind of traffic reliability for a freeway link consisting of several sections. The overload probability for a single bottleneck (either distinct or virtual, cf. Fig. 1) is equal to the capacity distribution function Fc(q) given in Eq. 1. The probability of no congestion Pfree(q) represents the complementary event: Pfree (q)
= l-F c (q)
(9)
For the analysis of n subsequent (quasi-) bottleneck sections, the probability of no congestion in the whole system is the product of the single probabilities for each section: Pfree,1+2+...+n(q,,q2v..,qn) = fl[l-F c , i (q i )]
(10)
where qi = demand at section i Fc,i(qO = capacity distribution function of section i By applying the Weibull distribution function to each section, this can be written as:
P & ee,, + 2 + ... + n(q,,q2,-,qn)=e lP ' a *>'
* J = e ""'"
(11)
The overload probability for the whole system, which is equal to the capacity distribution function, is:
140 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
-t—
F c , + 2 + . . . + n ( q 1 , q 2 , . . . , q J = l - e "'P'"
(12)
In the special case of the traffic demand being identical in all sections (qi = q2 = ... = qn = q), the overload probability for the system is:
Fc,1+2+...+n(q) = l - e
«"'
(13)
This is again a Weibull distribution with the scale parameter pi+2+...+„:
Eq. 10 through 14 are based on the assumption that breakdown events due to an overload at the bottleneck sections and thus the capacity distribution functions are statistically independent. This assumption seems to be reasonable if the length of each section is sufficiently large. However, further empirical research may be required to establish the degree to which this assumption is justified. The fact that the volume of traffic arriving at bottleneck section i is influenced by congestion incidents at preceding sections (i - 1, i - 2, ...), e.g. due to the capacity drop, is not relevant because in case of an overload of one section, the whole system is regarded as overloaded according to the applied definition of traffic reliability. For the same reason, the impact of a queue spillback spreading over several sections does not affect the results.
TRAFFIC EFFICIENCY Brilon (2000) has proposed to use the parameter E = q-v-T
(15)
where E q v T
= = = =
traffic efficiency volume travel velocity over an extended section of the freeway duration of the time period for analysis of flow
[veh km/h] [veh/h] [km/h] [h]
as a measure to characterize the efficiency of traffic flow on a freeway. This parameter describes the "production per time unit" of a freeway. The more veh km a freeway produces per hour, the greater the efficiency with which the potential of the existing infrastructure is exploited.
Reliability of freeway traffic flow
141
By applying the concept of random capacities, each volume q has to be combined with the corresponding probability of a breakdown. Brilon and Zurlinden (2003) have derived: Ee Xp (q D ) = [ q D - v - ( l - x ) + q d c - v d c - x ] - T
(16)
where Eexp(qD) = qo = v = q dc =
expected efficiency at a demand volume qo demand traffic volume average velocity in fluent traffic for q = qo queue discharge volume
Vdc T
= queue discharge velocity = duration of the period under investigation
X
=
1
h
[veh km/h] [veh/h] [km/h] [veh/h]
[km/h] [h]
m
(
li-k)-( 1 -Pco I ,g,i-k-l)
H i-l k-0
n m ricong pe(q)
= expected proportion of congested intervals with q = qdc = number of 5-minute intervals during T
[-] [-]
= Minjncong-l;i-1} = average duration (number of intervals) of a congested period >1 = probability of a breakdown at volume q (here: q = qo) (e.g. after Eq. 4: p B = dF c (q)/dq)
[-] [-] [-]
m
Pcong,i =
XPefai-lJ-O-Pcong.i-k-l) k=0
= probability of congested flow in 5-minute interval
[-]
If we insert real data for ps(q) = fc(q) into this set of equations it becomes clear that the maximum expected efficiency is achieved for a demand volume qo that is lower than the average capacity. Sample calculations show that the highest efficiency of a freeway is to be expected at a demand of approximately 0.9 c, where c is the traditionally defined (constantvalue) capacity.
CONCLUSIONS As a result of a series of studies of German freeways, the concept of stochastic capacities seems to be more realistic and more useful than the traditional use of single value capacities. This probabilistic approach provides an improved understanding of both the variability of traffic flow observations and the typical dynamics in different traffic states on a freeway. The idea of random capacity is based on the work of authors like van Toorenburg (1986) and Minderhoud et al. (1997). Compared to their approach, the so-called Product Limit Method for capacity estimation has been modified and extended. It was rather distinctly shown that the capacity of a freeway section is Weibull-distributed. For German freeways, the shape parameter seems to be in a range of 13, whereas the scale parameter can be different for
142 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction specific freeway sections. One drawback of the methodology is that huge sample sizes are needed to estimate capacity distribution functions. With the application of ITS-methods, these data will become increasingly available. The concept of randomness permits to demonstrate the capacity reducing effect of wet road surfaces (-11 %) and the capacity increasing effect of traffic adaptive variable speed limits. The Product Limit Method was also used to estimate queue discharge capacity. It was confirmed that this capacity is usually lower than the capacity in fluent traffic. The studies did clearly show that three typical states in traffic flow exist: fluent traffic, congested traffic, and a transient state that occurs in each breakdown and recovery of traffic flow. The extent of the so-called capacity drop did not show any regularities, although many freeway sections, each with a large sample size, have been analyzed. It is concluded that the capacity drop has rather chaotic properties. The concept of random capacity reveals that the optimum degree of saturation for a freeway, based on data from Germany, ranges around 90 %. If the degree of saturation increases further, the risk of a breakdown becomes too high, so that the efficiency of freeway operation must be expected to be lower than at a saturation of 90 %. The stochastic concept can also be applied to freeways consisting of several succeeding sections. Overall, it is expected that the random interpretation of freeway capacity offers the potential for improved traffic engineering methodologies.
REFERENCES van Aerde, M. (1995). A Single Regime Speed-Flow-Density Relationship for Freeways and Arterials. Proceedings of the 74th TRB Annual Meeting. Washington D.C. Banks, J. H. (1990). Flow Processes at a Freeway Bottleneck. Transportation Research Record No. 1287, Transportation Research Board, National Research Council, Washington D.C. Brilon, W. (2000). Traffic Flow Analysis beyond Traditional Methods. Proceedings of the 4th International Symposium on Highway Capacity, pp. 26-41, TRB Circular E-C018, Transportation Research Board, Washington D.C. Brilon, W. and H. Zurlinden (2003). Ueberlastungswahrscheinlichkeiten und Verkehrsleistung als Bemessungskriterium fuer Strassenverkehrsanlagen (Breakdown Probability and Traffic Efficiency as Design Criteria for Freeways). Forschung Strassenbau und StrassenverkehrstechnikNo. 870. Bonn. Elefteriadou, L., R.P. Roess, and W.R. McShane (1995). Probabilistic Nature of Breakdown at Freeway Merge Junctions. Transportation Research Record 1484, Transportation Research Board, National Research Council, Washington D.C. Hall, F.L. and K. Agyemang-Duah (1991). Freeway Capacity Drop and the Definition of Capacity. Transportation Research Record 1320, Transportation Research Board, National Research Council, Washington D.C. HCM (2000). Highway Capacity Manual. Transportation Research Board, Washington D.C.
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Kaplan, E.L. and P. Meier (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457-481. Keller, H. and T. Sachse (1992). Einfluss des Bezugsintervalls in Fundamentaldiagramrnen auf die zutreffende Beschreibung der Leistungsfaehigkeit von StraBenabschnitten (Influence of Aggregation Intervals in the Fundamental Diagram on the Appropriate Description of Roadway Capacity). Forschung Strassenbau und Strassenverkehrstechnik No. 614. Bonn. Kerner, B. S. (2000). Theory of Breakdown Phenomenon at Highway Bottlenecks. Transportation Research Record No. 1710, pp. 136-144. Transportation Research Board, National Research Council, Washington D.C. Kerner, B. S. and H. Rehborn (1996). Experimental Properties of Complexity in Traffic Flow. Physical Review E, 53 (5), 4275-4278. Kim, Y. and H. Keller (2001): Zur Dynamik zwischen Verkehrszustanden im Fundamentaldiagramm (Dynamics between Traffic States in the Fundamental Diagram). Strassenverkehrstechnik, Issue 9/2001, pp. 433-442. Kuehne, R.D. and N. Anstett (1999). Stochastic Methods for Analysis of Traffic Pattern Formation. Proceedings of the 14' International Symposium on Transportation and Traffic Theory. Jerusalem. Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data. Wiley, New York. Lorenz, M. and L. Elefteriadou (2000). A Probabilistic Approach to Defining Freeway Capacity and Breakdown. Proceedings of the 4th International Symposium on Highway Capacity, pp. 84-95. TRB-Circular E-C018, Transportation Research Board, Washington D.C. Minderhoud, M.M., H. Botma and P.H.L. Bovy (1997). Roadway Capacity Estimation Methods Explained and Assessed. Transportation Research Record No. 1572. Transportation Research Board, National Research Council, Washington D.C. Okamura, H., S. Watanabe and T. Watanabe (2000). An Empirical Study on the Capacity of Bottlenecks on the Basic Suburban Expressway Sections in Japan. Proc. of the 4th International Symposium on Highway Capacity, pp. 120-129. TRB Circular E-C018, Transportation Research Board, Washington D.C. Persaud, B., S. Yagar and R. Brownlee (1998). Exploration of the Breakdown Phenomenon in Freeway Traffic. Transportation Research Record No. 1634, pp. 64-69. Transportation Research Board, National Research Council, Washington D.C. Ponzlet, M. (1996). Dynamik der Leistungsfahigkeiten von Autobahnen (Dynamics of Freeway Capacity). Schriftenreihe des Lehrstuhls fuer Verkehrswesen der RuhrUniversitaet Bochum, No. 16. Bochum. Regler, M. (2004). Verkehrsablauf und Kapazitat auf Autobahnen (Freeway Traffic Flow and Capacity). Schriftenreihe des Lehrstuhls fuer Verkehrswesen der Ruhr-Universitaet Bochum, No. 28. Bochum. Shaw, T. (2003). Performance Measures of Operational Effectiveness for Highway Segments and Systems. NCHRP Synthesis 311, Transportation Research Board, Washington, D.C. van Toorenburg, J. (1986). Praktijwaarden voor de capaciteit. Rijkswaterstaat dienst Verkeerskunde, Rotterdam.
144 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Treiterer, J. and J.A. Myers, (1974). The Hysteresis Phenomenon in Traffic Flow. Proceedings of the 6' International Symposium on Transportation and Traffic Theory, pp. 13-38. Reed Pty Ltd., Sydney.
8
A CRITICAL COMPARISON OF THE KINEMATIC-WAVE MODEL WITH OBSERVATIONAL DATA Kai Nagel, Inst. For Land and Sea Transport Systems, Technical University of Berlin, Berlin, Germany; and Paul Nelson, Dept of Computer Science, Texas A&M University, College Station, Texas, USA
INTRODUCTION The objectives of this work are to: critically summarize the predictive capabilities and predictions of the classical kinematic-wave model (KWM) of Lighthill and Whitham (1955) and Richards (1956), where by "classical" is intended the case of a strictly concave fundamental diagram; critically summarize the extant data that are and are not reproduced by those predictions, especially with a view toward identifying the disagreements between observations and the classical KWM that seem most firmly established. Whether an allegedly observed phenomenon is "firmly established" is subjective. An element is reproduction by multiple independent observers; however, that is not sufficient, because it is possible for multiple observers to build upon the same mistaken assumption or technique. It is therefore possible for two reasonable people to disagree as to whether existence of an alleged phenomenon has been firmly established; indeed the two authors do not completely agree in all instances. Our approach in cases where either of us has doubts is simply to summarize, as objectively as we can manage, both sides of the issue. There also arise issues of interpretation of the predictive variables internal to a theory. This is especially relevant to continuum models, of which the KWM is an instance, because these models have continuous functions (e.g., mean speed) as their predictive variables, whereas any particular instance of traffic flow consists of discrete vehicles in motion. Tying theoretical continuum results to predictions is particularly difficult because much of the available data
146
Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
is obtained by aggregating over time at sparsely spaced detectors near ramps, yet there seems to be few studies of the effect of aggregation interval, detector spacing or detector location. The contents of this work are as follows. In the following section we summarize the predictions of the classical KWM, and associated issues, especially those involving computational employment of the KWM. The next section summarizes instances of agreement of the KWM with observations (qualitative features, cumulative flows), and provides a critique of such observations, including especially identification of areas where further observational studies seem warranted. The following section is devoted to a critical discussion of alleged observations that apparently disagree with the classical KWM. The specific alleged discrepancies discussed there include unstable flow and related phenomena (spontaneous breakdown, the two-capacity phenomenon), what we term empirical jammed flow (particularly "wide jams"), and internal structure in congested flow. Our final section contains conclusions and recommendations for further related work.
THE CLASSICAL KWM The classical KWM for traffic consists of the equation of continuity, d,p(x,t) + dxq{x,t) = 0, plus the assumption that flow (q=q(x,t)) can be written as a function (FD) of density (p=p(x,t)) and possibly explicitly of longitudinal position (x) and time (i), say q(x,t) = Q{p(x,t\x,t). Here Q is strictly concave in p, i.e. Q"0, there passed exactly one characteristic intersecting the x-axis (i.e., t=0), then the density at that point would be uniquely determined as that initially specified at that point of intersection with the initial line. There exist situations in which this stipulation holds, but there are two ways it can fail.
Critical comparison of the kinematic-wave model with observational data 147
Figure 1 - A strictly concave FD, as typically conceptualized in regard to the KWM. First suppose density is initially increasing as one moves downstream, at the initial time (e.g., near the end of a rush period). Because of the strict concavity the wave speed is (algebraically) decreasing along that interval, and therefore any two characteristics through distinct points along this roadway will eventually intersect. At such a point the "constant density along characteristics" specification of the preceding paragraph therefore fails to specify a unique value of the density. This ambiguity is traditionally resolved via a shock wave, that is a curve x=xs(t) in the (t,x)plane that is the locus of the initial points of intersection of characteristics, and along which the density (and hence flow) are permitted to be discontinuous. The shock condition dx, = Aq =Q(pu(t))-Q(pd(t)) „. dt
Ap'
pu(t)~PA')
then follows from conservation of vehicles. Here pu(t)=p(xs(t)-,t) (pd(t)=p(xs(t)+,t)) is the density immediately upstream (downstream) of the shock wave at time t. If one applies the "constant density along characteristics" specification only so long as the characteristic does not cross a shock, then the solution again is uniquely specified everywhere. In particular, Eq. (3) is an ordinary differential equation that determines the trajectory of the shock wave. In case the initial density decreases as one moves downstream no corresponding difficulty arises; the characteristics diverge ("fan out") with increasing time, but one and only one characteristic passes through each point (t,x), t>0. However, in the limiting case of a jump discontinuity in the initial density at some xo, with the downstream density smaller (e.g., at a traffic signal that has just turned green), there are points (t,x), t>0, through which no characteristic passes. In this case one mathematical (weak) solution is a shock, with densities downstream (upstream), up to the downstream (upstream) characteristic passing through (t,x)=(O,xo), equal to p(xg+,0) {p(xo-,O)), and trajectory otherwise determined by Eq. (3), just as in the preceding paragraph. However a second solution that fills the gap between the up-
148 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction stream characteristic through (t,x)=(0, xo) and the downstream characteristic through the same point is given by the similarity solution
p( x , 0 = c -^£zfij,
(4)
where c1 is the functional inverse of the wave velocity, which exists because of the strict concavity of Q. In case of such initial discontinuities one can create infinitely many (weak) solutions by piecing together segments of this similarity solution with pieces of the preceding shock solution. It is generally deemed that the similarity solution is that best representing the true state of traffic. The simplest mathematical argument supporting that choice is "stability": if one considers a sequence of initial densities having linear segments about x=xo that converge to the discontinuous initial density, and otherwise are equal to that initial density, then the corresponding (now uniquely determined, as previously) solutions resemble and converge to the similarity solution. That is, the similarity solution is self-healing, in that small perturbations from it return to it, whereas (in the present case) small perturbations of the shock solutions diverge from that shock solution. The simplest argument supporting the choice of similarity and involving driver behavior is that any shock segment represents some drivers traveling more slowly than they safely could, according to the prevailing FD. The choice of the similarity solution also relates mathematically to the so-called "entropy condition," which in turn also has ties to driver behavior. In its application to the classical KWM the entropy condition essentially asserts (Ansorge, 1990) that drivers accelerate as soon as safely possible and wait as long as safely possible to decelerate, where safety is denominated by adherence to the given FD. But the entropy condition is relevant mostly because it can be given a quantitative formulation (e.g., Leveque, 1992) that can be applied to computational approximations. This is extremely important, because early within the computational development of traffic flow theory some investigators mistakenly concluded that the classical KWM was fundamentally flawed because they inappropriately employed simple and reasonable computational approximations that failed to satisfy the entropy condition. The corresponding computational approximations, for an initial discontinuity as above, then converged with mesh refinement to the often clearly incorrect unstable shock solution, rather than to the stable similarity solution; cf. Ross (1988), Newell (1989), and Ross (1989). This mistaken conclusion seems to have provided much of the impetus for development of so-called higher-order methods (Payne, 1971). Some of these have recently been shown (Aw and Rascle, 2000) to be seriously flawed, in that they can develop negative (i.e., upstream!) flows, even though all flows are initially downstream (and all initial densities nonnegative and less than jam density). The STRADA code (Buisson, 1997), which has been widely applied in France, is based on a computational solution of classical KWMs via the Godunov method (Lebacque, 1996), which has a discretization error that is first order in both space and time. Computational methods having higher-order computational approximations exist (Bale et al, (2002), but seem not to have been explored in traffic flow, perhaps because lower-order methods have some advantages for the real-time demands of traffic control. It is necessary, for any computational implementation, to consider a system of some finite length, and therefore to incorporate appropriate boundary conditions. See Lebacque (to appear) for a discussion of boundary conditions, and Nelson and Kumar (2004) for the extension of these to interfaces (jump disconti-
Critical comparison of the kinematic-wave model with observational data 149 nuities of the FD in x) and point constrictions (removable singularities, in x, of the FD). In these cases consideration of only the similarity solution, for downward initial jumps in density as one moves downstream, is again conventionally adopted to obtain uniqueness.
EMPIRICAL AGREEMENT WITH THE CLASSICAL KWM In this section we review empirical results that appear to reflect agreement between observations and predictions from the classical KWM. The next section is devoted to a discussion of empirical observations that are often considered to reflect disagreement between observational data and the KWM. In many instances there are widely held but contrary opinions regarding these matters, for reasons discussed in the Introduction. The classification of a particular allegedly observed phenomenon as reflecting agreement or disagreement with the KWM is therefore somewhat arbitrary. Qualitative consistency between observations and the predictions of the KWM, particularly the shock waves and acceleration waves delineated in the preceding section, comprise perhaps the most widely accepted type of agreement. This is not easily or widely documented, perhaps because it is so widely accepted. For example, much of the technology underlying the Highway Capacity Manual, which is a semi-official U.S. standard for the analysis and design of roadway facilities, is based on supply-demand analysis, which is the steady-state version of the KWM. The situation is also complicated because many empirical studies that might provide verification have been reported in the literature related to a particular type of facility (e.g., freeway or signalized intersection), as opposed to the traffic modeling literature, and therefore not necessarily considered in light of consistency between data and the KWM. Shock waves Shock waves have been primarily observed in the context of freeway bottlenecks, which is to say freeway sections at which the capacity (maximum) flow in the FD of Fig. 1 is lower than immediately upstream. This is presumably because (inductive-loop) data from such locations are widely available, and because an understanding of the manner in which congestion at such locations forms and behaves is considered important to mitigation of the undesirable increase in travel times associated with congestion. According to the classical KWM, whenever the demand (flow) just upstream of the bottleneck entrance exceeds the capacity of the bottleneck, a shock wave will form at the entrance to the bottleneck and propagate upstream, leaving behind (i.e., bounded upstream by the moving shock discontinuity and downstream by the fixed bottleneck entrance) a "queue" of vehicles in which flow is equal to bottleneck capacity. However, in that queue the density will be the higher of the two densities corresponding to that flow on the FD, and therefore mean vehicle speed (= flow divided by density) will be (typically much) lower than that either upstream of the shock wave or downstream of the bottleneck entrance. The speed of the shock should be predictable from the shock condition (3). Agreements between these KWM predictions and observations have been documented, perhaps most notably in a series of data-oriented papers (Cassidy and Mauch, 2001; Windover and Cassidy, 2001) and companion modeling-oriented work (Newell, 2002) emanating from the transportation group at the University of California at Berkeley; see also Banks (1999).
150 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction This collective body of work even argues that a "zeroth order" (Newell, 2002) theory of highway traffic, which relies on an entirely triangular FD, is still able to predict accurately the most important effects of highway traffic, especially cumulative flows. Note however that cumulative flows are integral quantities, in the sense that at any location x they are the time integral of the differential flow q(x,t). Integral quantities are inherently easier to both measure and predict than the corresponding differential quantity; e.g., it appears that numerical differentiation, with the associated inherent loss of accuracy, will be required in order to obtain speeds and hence travel times from cumulative flows. The KWM characterization of a deceleration region as a "shock wave" of zero thickness is an idealization; nonetheless the actual structure of such transition zones upstream of queues has only recently been investigated empirically, by Mufioz and Daganzo (2003). These investigators describe detailed instances of shocks that are about 1 km wide, corresponding to a relatively gradual deceleration of about 1/3 m/sec2. This imposes some spatial (and presumably temporal) limitation on the scale of validity of the KWM, but these researchers further argue that even with such a wide shock the KWM can predict many relevant quantities quite accurately. As an example they cite vehicle positions to within 160 m, or about five vehicle spacings; however this bears further elaboration, because neither the KWM nor any other continuum model predicts vehicle spacing per se. Acceleration waves The focus in validation of the KWM has tended toward shock waves, perhaps because of the ready availability of data from "point" (e.g., inductive-loop) detectors on freeways, and the practical interest in congestion on freeways. There seems to be little observational data exemplifying the companion KWM prediction of acceleration waves. This is presumably because freeway detectors tend to be located upstream of potential bottlenecks, and therefore not in position to acquire data representative of the "queue discharge" that is the manifestation of acceleration waves downstream of a bottleneck. On the other hand, acceleration waves are extremely important to the behavior of traffic at signalized intersections; we believe the literature related to such intersections is an important source of potential data relative to validation of the KWM that has been too much neglected. A full realization of the potential use of such data would require spatially extensive measurements of flow during a green cycle, both upstream and downstream of the signal. With more widespread use of camera-controlled signals, such data seem destined to become more widely available. Even now some data are available, in the form of measurements of the "saturation flow" at a signal during the green phase; this parameter is extremely important to the performance of signalized intersections, and has been extensively discussed in the literature on signalized intersections. The prediction of the KWM is that (unopposed) queue discharges at such an intersection will be equal to the minimum of the capacity (maximum) flows immediately upstream of the intersection, at the intersection itself, or immediately downstream of the intersection, but in any event that it will remain constant during the green phase of the signal. In regard to this prediction, Lin, Tseng and Su (2004) state that "actual queue discharge patterns, however, often do not display an identifiable steady maximum rate." On the other hand, the variations in observed flow at the intersection that are reported by these investigators are only of the order of 10-15% (see also Bonneson, 1991). Thus here the KWM seems
Critical comparison of the kinematic-wave model with observational data 151 to be categorically neither a success nor a failure, but rather the appropriate view is dependent upon the level of accuracy required. Lebacque (2002) has considered replacing the entropy condition for the KWM by a "bounded acceleration" condition, because of doubts about accuracy of the entropy solutions for modeling ramp metering. It should be noted that a KWM with a strictly concave FD does not predict a constant flow rate out of the traffic light, except exactly at the position of the light. Boundary-induced breakdown The preceding section mentions the FD as a hypothetical representation of flow as a function of density. Let us call the FD so related to the KWM theory KWM-FD. One can also measure flows and densities in the field, leading to an empirical FD. Classically such observations lead to empirical FDs that resemble the KWM-FD well in the lower-density free-flow regime, but at higher densities display a tail extrapolating the free-flow data lying above a "cloud" of widely scattered data points (Drake, Schofer and May, 1967, Koshi, Iwasaki and Ohkura, 1981; Kerner, 2002, esp. Figs. 14 and 16); cf. Fig. 2 for a typical instance of such an "inverted-lambda" empirical FD. Often it is better to plot density p, flow q and velocity v as functions of time, as in Fig. 3. These time-series data tend to show that a typical transition is from the free-flow regime to a regime where flow is only somewhat diminished but densities are much higher, meaning much lower velocities (Mika, Keer and Yuan, 1969; Kerner and Rehborn, 1996a; Kerner, 1998, 1999a); such a transition often is termed "breakdown."
figure z - empirical iunaameniai diagram, as recoraea on me uerman ireeway A m. uensny (on the x axis) is given in veh/(km lane), flow (on the y axis) is given in veh/(hour lane). Source: Nagel, Wagner, and Woesler, 2003. Daganzo, Cassidy and Bertini (1999) note that a bottleneck downstream of a measurement location can easily generate such a plot, in the following way: The system starts with low flow at low densities. Both flow and density keep increasing, along the "free flow" branch of the fundamental diagram.
152 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction This flow can be larger than can flow through the bottleneck. A queue thus begins to form upstream of the bottleneck, but that does not immediately influence the measurement. Eventually, the queue will reach the measurement location. At that point in time, data points will move to a much higher density, while the flow value will now drop to bottleneck capacity. This is exactly what the KWM predicts, in that there is a transition from free flow to congested flow that accompanies a shock wave propagating upstream. Thus this must be marked as a success of the classical KWM. Unfortunately, there are too many flow-density data in the literature where the associated geometrical constraints are not well enough documented. Such data are relatively useless for many of the points raised in the current scientific discussion. At the same time alternative descriptions, such as that involving three phases, free flow, congested or synchronized and jammed "synchronized flow," do not yet seem to have evolved to the point that they can provide quantitative predictions even capable of being tested. (But see Schonhof and Helbing, submitted, and also Kerner, 2004.) This KWM interpretation of empirical observations of breakdown has recently (Cassidy, 1998) given rise to questions regarding the data analysis techniques traditionally employed for empirical FDs. Specifically, the empirical FD is obtained from averaging, for example over fixed time intervals. If one assumes that the KWM describes traffic, then these averages can extend over several regimes. This means that linear combinations of points on the KWM-FD can become points of the empirical FD. For a concave KWM-FD this means that points within the KWM-FD can become an empirical data point.
Figure 3 - Time series of the three fundamental variables of traffic flow. Flow and velocity are averaged; density is computed as *, where > denotes the mean. The data have been recorded on the German freeway Al near an intersection with German freeway A59. Time is counted from 6/Jun/1996 midnight. Source: Nagel, Wagner, and Woesler, 2003.
Critical comparison of the kinematic-wave model with observational data 153 Cassidy and Bertini (1999) pursue this line of reasoning in some detail. They also describe how averages can be restricted to time intervals of (nearly) stationary traffic, ensuring that the empirical data points lie on a possibly existing KWM-FD as much as possible. Unfortunately, these results are not conclusive as regards the possibility of obtaining a KWM-FD via filtering empirical data for stationarity, because data so filtered are sparse both near capacity flow and in the congested flow regime. On the other hand, such sparsity is exactly what would be predicted by the KWM, in the region upstream of a bottleneck, which is where data customarily are taken (and indeed where the KWM predicts data must be taken, in order to observe any congested data). That is, the KWM predicts that once flow at an observation point upstream of a bottleneck exceeds the capacity of that bottleneck, it will do so for only a limited time, after which the queue formed at the bottleneck will spill back to the observation point and flow at that point subsequently will be equal to the bottleneck capacity (i.e., will correspond to a single density-flow point). Indeed, the chief inconsistency between the filtered nearstationary data and the KWM seems to be the existence of multiple data points in the congested regime (cf. Fig. 11 of Cassidy and Bertini, 1999). We believe much remains to be done in the area of data analysis related to empirical FDs.
EMPIRICAL DISAGREEMENT WITH THE CLASSICAL KWM In this section we review empirical results that seem to reflect disagreement between observations and predictions from the classical KWM. As before the difference between "agreement" and "disagreement" is considerably subjective. Further, the disposition toward perceiving consistency or inconsistency between the KWM and a particular set of observations often is reflective of familiarity with the KWM or some alternative theory. We make an effort to factor out such inherent bias, but perfection in that regard is elusive. Unstable flow, spontaneous breakdown and the two-capacity phenomenon In the preceding section we have already offered one hypothesis that might somewhat reconcile data such as Fig. 2 with the KWM-FD, but if one takes such data at face value - as has been done through much of the history of traffic flow theory - then it seems doubtful they can be represented adequately by any single continuous function. This observation is classical in traffic flow theory, and is behind (e.g.) the well-known "two-regime" theory (Ceder, 1967). The interpretation of this "scatter" regime is controversial, and even the terms one chooses to employ in referring to it can evoke bias toward one or another interpretation, and perhaps associated theory. In an effort to avoid such bias, we employ the following terminology: By empirically congested flow we shall intend any data clearly not falling within the free flow regime, as defined above. By empirically uncorrelatedflow we intend any data set within which there is considerable scatter in the density-flow plane, as illustrated in Figure 2. (Thus empirically uncorrelated flow is a subclass of empirical congested flow.)
154 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction By empirically jammed flow we shall intend yet another subclass of empirically congested flow that has been hypothesized by some investigators, as discussed in more detail in the following subsection. The word "empirical" and its derivatives will be omitted when the context permits. The traditional acceptance of these density-flow measurements in the (empirically) congested regime raises doubts regarding the validity of the KWM. This alone has been persuasive to many, and has fueled both substantial skepticism directed toward the KWM, and a search for alternative data interpretations and associated explanatory theories. On the other hand, the successes of the KWM discussed in the preceding section have made others considerably reluctant to accept any suggestion that this theory has no validity at higher densities. In the preceding section we have already indicated a recently suggested possibility for reconciling kinematic-wave theory with empirically uncorrelated flow. The fundamental hypothesis underlying that possibility is that breakdown is exclusively what nonlinear dynamicists would term a "boundary-driven phenomenon," which is to say that it derives from shocks propagating upstream from a bottleneck, as summarized from Daganzo, Cassidy and Bertini (1999) in the preceding section. In our opinion such bottleneck-driven breakdowns certainly occur, and empirical reports should include sufficient information to permit determination of the extent to which boundary effects contribute to data reflecting breakdown. This is a nontrivial challenge, because it requires spatially distributed data, or at least a very careful description of the roadway downstream of the point of observation. An extreme version of this suggestion is that empirical results should not be published at all, unless the data on which they are based are made publicly available, so that alternative hypotheses can also be tested against those data. Nonetheless, when one takes into account the possibility of bottleneck-driven breakdown, there remain observed phenomena that are difficult to explain absent spontaneous ("emergent") breakdown. The objective of this subsection is to summarize the observational data supporting such spontaneous breakdown. More generally, bottleneck-driven breakdown is an instance of boundary-driven phenomenon, which in turn is instance of an "external" phenomenon that stems from circumstances outside the system being modeled, while spontaneous breakdown is conceived as an emergent phenomenon that arises from phenomena internal to the system being modeled. However, to a very large extent this distinction is subjective, in that it depends upon what one is willing to assume is known a priori (i.e., an external variable), as opposed to being a quantity that the model is required to predict (i.e., an internal variable). As a very concrete instance of this dichotomy, the presence of a slow truck in the right lane certainly can be a significant factor in determining traffic flow. But should this presence be regarded as an external factor, and therefore known a priori, or as a stochastic fluctuation whose potential existence must be represented among the statistical distribution giving rise to the mean values that a continuum model should provide? There exist circumstances under which either of these views is reasonable, depending upon the purpose of the model, and therefore what one is willing to assume is known a priori. For example, if the objective of the model is to better understand the effect of a slow truck (e.g.,
Critical comparison of the kinematic-wave model with observational data 155 Daganzo and Laval, 2005), then it is perfectly appropriate to take the presence (and characteristics) of such a truck as a known external variable. However, if the purpose is to predict mean traffic behavior where it cannot be (continuously) measured then the presence or absence, characteristics, and effect of such a truck must be regarded as a stochastic internal variable to be appropriately reflected in the model predictions. The simplest way to determine internal effects for a model of traffic flow is to implement it on a homogeneous closed loop, which completely eliminates external (boundary) effects. Such "ring roads" are rare in practice, but Sugiyama et al. (to appear) recently reported a relevant experiment conducted on a test circuit. Briefly, they experimentally show that under appropriate circumstances, spatially homogeneous flow is unstable for sufficiently high densities, but rather tends "spontaneously" to transition, after several minutes, to "jammed flow." In this context the characteristic of jammed flow is a single jam, propagating opposite to the direction of traffic flow, within which vehicles come to a stop, or very nearly so. Quantitative details of the observed speed-headway distribution over the entire jam flow are not reported, but supporting simulations suggest they comprise a characteristic ("universal") limit cycle ("hysteresis loop"). This result suggests that spontaneous breakdown can occur in traffic flow. However, in itself it signifies neither failure of the KWM nor nonexistence of a KWM-FD, in the sense that flows and densities averaged over the characteristic limit cycles could still comprise a perfectly well defined function in the density-flow plane. This possibility is illustrated by the cellular automata model CA-184a of Nelson (submitted), which produces results similar to the empirical FD of Fig. 2, in simulations of traffic stream observations upstream of a bottleneck, but that on a ring road fall exactly on a well-defined KWM-FD in the density-flow plane, with multiple underlying limit cycles. However, at a minimum the results of Sugiyama et al. (to appear) suggest existence of a lower limit on the spatial and temporal scales for validity of the KWM; existence of some such scale limitations is to be expected for any continuum model; cf. Lesort et al. (to appear) for further discussion of scale issues in traffic flow. In addition, Jost and Nagel (2003) argue that if the spontaneous breakdown is related to a true (thermodynamical) phase transition, then the jam features will coarsen until they are visible on all spatial scales, even the largest. The mechanism for coarsening is that jams coagulate until there is eventually only one large jam left in the system. However, they cannot answer if their modeling results apply to real-world traffic or not. Observations of what we term as "two-capacity flow" have been persistently reported in the literature. This phenomenon is relevant here because it also is strongly suggestive of spontaneous breakdown, and is inconsistent with the classical KWM. Here by two-capacity flow we mean observations and related analyses that suggest bottlenecks can operate at two distinct maximum flows, typically a higher value prior to and early in the formation of a queue upstream of the bottleneck and a somewhat lower value subsequent to development of such a queue. This contradicts the classical KWM, because the (entropy solution of the) latter predicts flow at an enqueued bottleneck will be a constant, specifically the capacity in the bottleneck (Nelson and Kumar, 2004). Observations of two-capacity flow have been reported, among others, by Edie and Foote (1960), by Agyemang-Duah and Hall (1991) and by Cassidy and Bertini (1999), while other
156 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction studies (e.g., Persaud and Hurdle, 1991) have reported lack of conclusive evidence supporting a reduction in flow coincident with formation of a queue. From an empirical perspective, Banks (1991; cf. also Banks, 1990 and Banks, 1991a) indicates that "the hypothesis that flow decreases when it breaks down is confirmed, provided the hypothesis applies to individual lanes," but "when averaged across all lanes there was no significant change," and concludes that alleged two-capacity flow "is unlikely to provide a basis for metering." Note that if one fully subscribes to the KWM, then there is no rational basis for ramp metering, absent some societal decision to favor mainline traffic over that entering freeways. When differences between the levels of flow at a bottleneck, before and after queue formation, are reported, their magnitude tends to be only about 5-10%. Given that relatively small alleged difference, it is difficult to know whether the different conclusions reached in different studies are attributable to differences in methodologies for obtaining and analyzing data, or to actual differences in traffic behavior at distinct sites, or possibly some combination of these factors. This inherent difficulty in determining existence of the two-capacity phenomenon has been discussed by Persaud and Hurdle (1991). Nonetheless one can question whether we have yet advanced significantly beyond the status observed by Wattleworth (1963): "The question of whether or not the flow downstream of a freeway bottleneck decreases when congestion sets in is currently the subject of much discussion in engineering circles. Research findings support both the yes and no answers to this question. Several studies ... suggested that perhaps the question did not have a simple yes or no answer." Yet, on balance we tend to agree with Cassidy and Bertini (1999) that "the average rate vehicles discharge from a queue can be ... lower than the flow measured prior to the queue's formation." See Elefteriadou, Roess and McShane (1994), and Lorenz and Elefteriadou (2001) for a suggestion that the time of occurrence of this type of breakdown can usefully be viewed as stochastic in nature.
Empirically jammed flow Some researchers perceive an additional subclass of empirically congested flow, which is the empirically jammed flow already mentioned above. The characteristic signature of empirically jammed flow is so-called "wide jams," which are described by Knospe et al. (2002) as follows: "Wide (moving) jams are regions with a very high density and negligible average velocity and flow. The width of these structures is much larger than its fronts at the upstream and downstream ends where the speed of the vehicle changes sharply. Wide jams can propagate undisturbed through either free flow or synchronized traffic without impact on these states, which thus allows their co-existence." (See the following subsection for more on "synchronized flow.") The predominant evidence for wide jams is time-series data such as that shown in Fig. 23 of Knospe et al. (2002); i.e., data from a stationary (loop) detector that show at early times relatively large flows and speeds with small densities, followed by an intermediate period of small flows and speeds and high densities, finally followed by yet a third time period qualitatively resembling the first. The hypothesis commonly invoked is that such data represent a structure ("wide jam") traveling upstream, within which vehicles are closely spaced, and in which both upstream and downstream fronts remain sharply defined as the structure propagates. If this interpretation is correct, then it provides a challenge to the classical KWM, because (see the section entitled "The Classical KWM") that model predicts that an initially
Critical comparison of the kinematic-wave model with observational data 157 sharp front between a high-density upstream region and a low-density downstream region (such as the downstream front of the hypothesized wide jam) will dissipate (spread) into an acceleration fan (similarity solution). However, absent additional data from other spatial points there is an interpretation of these data that reflects a boundary-driven phenomenon, and is consistent with the classical KWM. What is required is the sequence of events suggested in the preceding section as providing an explanation of Fig. 3 consistent with the KWM, followed by a decrease of upstream demand to a value again below bottleneck capacity. The KWM then predicts a sequence of events as already described, until the upstream demand falls below the bottleneck capacity. Once that decreased demand reaches the tail of the queue the tail will remain a shock wave, and thus be a stable structure within the confines of the classical KWM, but now it will propagate downstream. It will eventually reach the detector, where it will be registered as a sharp transition from a high-density low-flow/speed regime to low-density high-flow/speed. As seen at the hypothesized single detector the traffic pattern will thus be exactly the observed signature of an alleged wide jam, as described in the preceding paragraph. Thus the question of whether the data supporting existence of wide jams is consistent with the classical KWM comes down to the whether the second observed sharp front is moving upstream or downstream. This seems to require data from more than one spatial location. At this time the totality of the data of the type described above that has been reported in the literature in support of the wide-jam hypothesis seems to be Kerner and Rehborn (1996a, 1996b), Knospe et al. (2002) and Schonhof and Helbing (submitted). Of these three works, only in Figures 3(a)-(c) of Kerner and Rehbom (1996a) and Figure 2 of Kerner and Rehborn (1996b) does one find explicit measurements from detectors at distinct positions along the roadway. These data definitely support existence of wide jams. It would be desirable if they were replicated by other investigators. Nonetheless, in the following sections we conditionally accept the hypothesis of existence of wide jams, and hence empirically jammed flow, and explore the potential for various models to reproduce that phenomenon. Some empirically properties of wide jams that have been reported (Kerner and Rehborn, 1996a, 1997) include the following: 1. Traffic jams, once created, are fairly stable and can move without major changes in their form for several hours against the flow of traffic. 2. The flow out of a jam is a stable, reproducible quantity, and can be up to 50% smaller than the maximal flow at the same location. Further internal structure in empirically uncorrelated flow The "wide jams" characteristic of the empirically jammed flow discussed in the preceding subsection constitute one instance of internal structure in empirically uncongested flow. However, other types of internal structure have been reported. In fact, "stop-and-go" flow under congested conditions is anecdotally familiar to virtually all drivers. Notwithstanding that this phenomenon was initially studied scientifically nearly forty years past (Edie and Baverez, 1967), there remains a paucity of related quantitatively reliable scientific data. (But
158 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction see Schonhof and Helbing, submitted, for study of start-stop waves via a novel filtering technique applied to data from multiple dual-loop detectors.) This may be because such data require short-term and short-distance temporally and spatially distributed data, which necessarily will be subject to a high degree of stochastic uncertainty (stemming from, e.g., variations between individual drivers). Nonetheless there are persistent reports of "something else" (e.g., Kerner, Klenov and Wolf, 2002), and there is considerable discussion (Daganzo, Cassidy and Bertini, 1999) as to whether these observations are or are not entirely explainable by KWM theory. This subsection is devoted to a brief review of these issues. Much of the discussion focuses on what Kerner and collaborators (Kerner and Rehborn, 1996b, 1997) term as "synchronized flow." One of the barriers to an objective discussion of synchronized flow is the lack of consensus as to its characteristic signature(s). An approximately common (i.e., "synchronized") speed across multiple lanes, and the existence of embedded very narrow (compared to the wide jams of the preceding subsection) jams are two signatures that have been mentioned by investigators who have considered the matter. Yet another barrier is lack of public availability of much of the data on which many of the alleged observations of synchronized flow are based, and the consequent inability of the community at large to reproduce the underlying data analysis, or suggest alternate analyses. Even among those who are disposed favorably toward existence of synchronized flow there is not agreement on the question of whether the associated strong scatter of the data has a dynamical origin, for example stemming from an on-ramp, or a statistical origin, such as being caused by a mixture of cars and trucks that display different driving characteristics. Sugiyama et al. (to appear) provide simulation results, based on the optimal-velocity model (Bando et al., 1995) that suggest a hypothesis regarding possible internal structure of congested flow. Briefly, their simulations reveal the following structure upstream of a bottleneck: immediately upstream a region of "laminar flow," within which vehicles are spaced rather regularly, followed by a region within which the flow consists of very narrow "jams," finally followed by a region even further downstream within which wide jams appear. These researchers hypothesize the laminar region is an inherently metastable region, the region of wide jams is inherently stable, and the region of narrow jams is a connecting unstable region (as dictated by Hopf bifurcation) that is somehow stabilized by presence of the bottleneck. Note that the latter interpretation also is consistent with considerable difficulty in observing the narrow jams, which in turn would be somewhat consistent with difficulties in observing synchronized flow, if one hypothesized an identification between synchronized flow and narrow jams, as do Sugiyama and Nakayama (2003). The above description, in particular that narrow jams coagulate to form wide jams, is also consistent with work by Jost and Nagel (2003). Jost and Nagel do not investigate bottlenecks, but rather use a closed homogeneous system, i.e. a homogeneous loop. Once of their initial conditions is a completely homogeneous distribution of cars, i.e. all vehicles with the same space headway, as given by the density. Also in that system, one obtains, for certain densities, initially "laminar flow,", then "narrow jams," and then wide jams. As pointed out earlier, the wide jams will further coagulate until there is only one jam left in the system, and that statement holds for arbitrarily large systems. Further, unpublished simulations of our own show that the same code, after the introduction of a bottleneck, displays the same behavior as the model of Sugiyama et al. (to appear). An interpretation is that, either by a bottleneck or by having a closed system, traffic can be "pinched" at a certain (average) density, at which the
Critical comparison of the kinematic-wave model with observational data 159 system displays the above-described behavior. However, even here it must be said that there are very few empirical observations that include the spatial picture, and at least one of them (Windover and Cassidy, 2001) displays little or no coarsening with increasing distance from the bottleneck. The principal point of interest here is the extent to which structure internal to empirically congested flow disagrees with the KWM. Continuum models of traffic flow make predictions regarding density, mean speed and flow, all of which are continuum quantities that can be interpreted as mean values over some underlying distribution of vehicular speeds. But continuum models themselves are silent on the form of the associated speed distributions, and indeed regarding the nature of these distributions (i.e., over time and space, in a particular instance, or over an "ensemble" of instances). From this perspective structure within empirically congested flow perhaps provides information (e.g., the appropriate nature of the associated speed distribution) beyond that available from the KWM itself, but that information is not necessarily in contradiction to the KWM. See Nelson (submitted) for an example of a simple model of traffic flow that has such an internal structure within its congested regime, but nonetheless can be modeled reasonably well via the KWM, with an appropriately constituted FD. Note also the recent work of Bagnatini and Rascle (2003) in which a macroscopic model is displayed that displays spatial structure in its solutions, but has the property that suitably homogenized solutions nonetheless are solutions of the KWM. However, once more note that in some models structures can coarsen beyond any arbitrarily large but fixed length scale.
NONCLASSICAL KINEMATIC-WAVE MODELS A classical (strictly convex) FD will not reproduce the observed stable wide jams from a KWM. This section is a brief summary of what is known or believed about the ability of nonclassical KWMs to reproduce wide jams. Once we allow strictly linear pieces in Q(p), then some sort of stability of both fronts can be achieved, even for entropy solutions of the KWM, as long as both the density within the jam and the surrounding densities are within the same linear segment (Lin and Lo, 2003). This is because the slope of Q(p) denotes the phase velocity of the wave features, and if all densities in the range of interest are within the same linear segment, then their wave features move with the same velocity. Thus at least some of the elements of wide jams seemingly can be reproduced within the context of the simple triangular FD suggested by Newell (2002). However, such solutions seem to be only "weakly stable," in the sense that if the initial densities are slightly "smeared," then the ensuing density profile retains this smeared form, rather than reorganizing into a sharp front (i.e., displaying "self healing"). The bounded-acceleration extension (modification?) of the KWM (Lebacque, 2002) does not require the FD to hold in circumstances under which this would require maximum accelerations that exceed some a priori specified bound, but rather replaces the FD by the stipulation that the maximum acceleration occurring is equal to that bound. This modification has been demonstrated to lead to stable wide jams; see the reference cited for details. It is, however, a matter of taste if a model with bounded acceleration still is a kinematic model in
160 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the strict sense, since bounded acceleration actually means that flow is no longer a function of local density only. hi unpublished work the authors have considered the KWM under an FD with a strictly convex but decreasing tail (i.e., positive second derivative and negative first derivative). The corresponding entropy solutions of the Riemann problem can be worked out, using results from the mathematical literature (Li, 2003, and citations therein). These solutions have some interesting features (e.g., deceleration fans and acceleration shocks). These can be explained in terms of hypothesized driver behavior. However, it does not appear that this approach is capable of leading to stable wide jams.
CONCLUSIONS This paper was written with the intent to first reach a firm basis of agreement between the two authors, and then to continue from there with the description and analysis of simulations that describe some of the more subtle effects of traffic flow, such as structure formation in queues. Somewhat surprisingly, just reaching the firm basis has exhausted the limits given for this paper. This seems to be caused by the following: (1) Both authors have their own intuitions about the dynamics of traffic flow. (2) Both authors agree that the other author's intuition is valid, although considerable explanation and thoughtfulness was necessary in order to have that agreement based on true mutual understanding. (3) Both authors' intuitions are consistent with the data sets that they were aware of; indeed, only very few data sets are able to answer at least some of the critical questions. The perhaps most significant example is the breakdown experiment of Sugiyama et al. (to appear), which, although it has been reported in 2001, has neither been published not been widely disseminated. Nevertheless, it is the only unequivocal example of spontaneous breakdown that we are aware of. Similar statements hold for the availability of data that are both temporal and spatial. The most important results of our effort are the following: The kinematic wave model (KWM) explains and predicts many if not most effects of real world traffic dynamics. As pointed out, e.g., by Daganzo, Cassidy, and Bertini (1999), much of the single detector data supposedly in support of spontaneous breakdown can also be explained by queue dynamics in conjunction with a geometrical constraint. On the other hand, there are some data sets that are, in both authors' views, very difficult to reconcile with the traditional KWM. This holds certainly for KWMs with strictly concave fundamental diagrams (FDs), which do not explain stable jams, spontaneous breakdown out of nothing, or structure formation in queues. Structure formation in queues could be argued as being beyond the spatial scale that KWMs claim to deal with, although a "true" phase transition interpretation of traffic flow would predict that these structures coagulate into larger and larger structures with increasing distance from the bottleneck. Finally, there seems to be only one unequivocal observation of spontaneous breakdown. As said before, more traffic data that include the spatial aspect is needed in order to make further progress. Further, the field of traffic-flow theory and analysis would benefit from an international effort to provide universal access to data supporting published analy-
Critical comparison of the kinematic-wave model with observational data 161 ses, and support from journal editors to make provision of such data to this project a prerequisite to publication would be helpful.
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Critical comparison of the kinematic-wave model with observational data 163 Lighthill, M. J. and G. B. Whitham (1955). On kinematic waves II - a theory of traffic flow on long crowded roads. Proc. Royal Society, London, A229, 317-345. Li, T. (2003). Global solutions of nonconcave hyperbolic laws with relaxation arising from traffic flow. Journal of Differential Equations, 190, 131-149. Lin, F. B., P. Y. Tseng and C. W. Su (2004). Variations in queue discharge patterns and their implications in analysis of signalized intersections. In Proc. 83rd Annual Meeting Transp. Research Board, Wash. D.C., Jan. 11-15. Lin, W.-H. and H. K. Lo. A theoretical probe of a German experiment on stationary moving traffic jams. Transp. Res. B, 37, 251-261. Lorenz, M. R. and L. Elefteriadou (2001). Defining freeway capacity as function of breakdown probability. Transp. Research Record, 1776, 43-51. Mika, H. S., J. B. Keer and L. S. Yuan (1969). Dual mode behavior of freeway traffic. Highway Research Record, 279, 1-13. Mufioz, J. C. and C. F. Daganzo (2003). Structure of the transition zone behind freeway queues. Transp. Science, 37, 312-329. Nagel, K., P. Wagner, and R. Woesler (2003), Still flowing: Approaches to traffic flow and traffic jam modeling. Operations Research, 51, 681-710. Nelson, P. (submitted). On two-regime flow, fundamental diagrams and kinematic-wave theory. Submitted to Transp. Science, March 2004. Nelson, P. and N. Kumar (2004). Point-constriction, interface and boundary conditions for the kinematic-wave model: In Proc. 83rd Annual Meeting Transp. Research Board, Wash. D. C , Jan. 11-15. Newell, G. F. (1989). Comments on traffic dynamics. Transp. Research B, 23B, 386-389. Newell, G. F. (2002). A simplified car-following model: A lower order model. Transp. Research £, 36B 195-205. Payne, H. J. (1971). Models of freeway traffic and control. Simulation Council Procs., 1, Simulation Council, Inc., La Jolla, CA, 51-61. Persaud, B. N. and V. F. Hurdle (1991). Freeway capacity: definition and measurement issues. In: Proc. Int. Symp. Highway Capacity and Level of Service (U. Brannolte, ed.), pp. 289-307, A. A. Balkema Press, Rotterdam. Richards, P. I. (1956). Shockwaves on the highway. Operations Research, 4, 42-51. Ross, P. (1988). Traffic dynamics. Transp. Research B, 22B, 421-435. Ross, P. (1989). Response to Newell. Transp. Research 5, 23B, 390-391. Schonhof, M. and D. Helbing (submitted). Empirical features of congested traffic states and their implications for traffic modeling. See www.helbing.org. Sugiyama, Y. et al. (to appear). Observation, theory and experiment for freeway traffic as physics of many-body system, hi: Traffic and Granular Flow '03. Sugiyama, S. and A. Nakayama (2003). Understanding "synchronized flow" by optimal velocity model, hi: Modeling of Complex Systems: Seventh Granada Lectures (P. L. Garrido and J. Marro, eds.), pp. 111-115, American Institute of Physics. Wattleworth, J. A. (1963). Some aspects of macroscopic freeway traffic flow theory. Traffic Engineering, A35, 15-20. Windover, J. R. and M. J. Cassidy (2001). Some observed details for freeway traffic evolution. Transp. Research A, A35, 881-894.
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AVERAGE VELOCITY OF WAVES PROPAGATING THROUGH CONGESTED FREEWAY TRAFFIC Benjamin A. Coifman, Department of Civil and Environmental Engineering and Geodetic Science, and Yun Wang, Department of Electrical and Computer Engineering, Ohio State University, Columbus, Ohio, USA
INTRODUCTION Large portions of traffic flow theory are built upon the bivariate relationship between flow and density or the related relationship between headway and spacing. This scope begins with the first-order, macroscopic model of Lighthill, Whitham and Richards (LWR) to predict the propagation of signals or waves through the traffic stream (Lighthill and Whitham, 1955; Richards, 1956) and continues into higher-order macroscopic models and microscopic models that attempt to model the behavior of individual drivers. But because it is difficult to measure density and uncommon to find stationary conditions during congested periods, debate continues about the shape of the bivariate curve and thus, implicitly impacts the models predicated upon it. It is widely agreed among traffic flow theorists that during free flow conditions flow increases monotonically with density, with a slight drop in slope as conditions approach capacity. But little can be said for certain about the congested regime beyond the fact that in aggregate, flow drops with increasing density. Empirical studies by Hall et al. (1986, 1988) propose an inverted V shape for the flow-occupancy curve, while noting that the findings were consistent with earlier studies. A few years later, out of convenience, Newell (1993) also chose to simplify the relationships by using a triangular flow-density curve for illustrative purposes in the course of theoretical development. Given a triangular curve, LWR predicts that during congested periods, waves should propagate upstream with a constant velocity, independent of
166 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the given flow, density, or local traffic speed.1 Subsequent empirical analysis suggests that indeed, during congestion, waves do appear propagate upstream with roughly constant velocity. Windover and Cassidy (2001) showed that the upstream moving waves could be found by comparing the cumulative arrival curves across all lanes between two neighboring loop detector stations. They used data from a 2 km segment of northbound Interstate-880, just south of Oakland, California, and found that the wave velocities during congested periods were nearly constant (17-20 km/h) over the homogeneous freeway segment, independent of flow. Mauch and Cassidy (2002) observed traffic data over multiple days along a 10 km stretch of the Queen Elizabeth Way (Ontario, Canada). They measured wave velocities by plotting the cumulative arrival curve across all the lanes from many successive detector stations with vertical displacements in proportion to the physical distance separating the successive detector stations and found that the waves propagating upstream in congestion had nearly constant velocities in the range of 22-24 km/h, independent of the location and the flow. Smilowitz and Daganzo (2002) employed the LWR model to derive the wave velocity from the slope of the line passing through the two states in accumulation-flow relationship, where the accumulation is proportional to the density between two observers. They found wave velocities on a single lane, homogeneous highway segment were 17.2 km/h on one day and 18.8 km/h on another. Deviating from the earlier techniques of manually extracting wave velocities, Munoz and Daganzo (2001) applied cross correlation to cumulative arrival curves summed across selected lanes for a single sample period and found a wave velocity of 19.4 km/h. These earlier efforts suggest that the flow-density curve is indeed triangular, e.g., Windover and Cassidy (2001) note, "traffic on homogeneous freeway segments might better be modelled using linear (triangular-shaped) relations." But all of the earlier studies used small samples, often aggregated across several lanes, to find the velocity of waves during congestion. If the wave velocities are roughly constant, independent of traffic conditions, they can be used as inputs to estimate link travel times (Coifman, 2002), which is part of the motivation behind this paper. To provide a more comprehensive analysis of wave velocity during congested periods, this research builds off of the cross correlation idea presented by Munoz and Daganzo (2001) and seeks to extract the average wave velocity automatically for each sample. Our methodology can accommodate additional data with little human intervention, thereby allowing the study of much larger data sets to better model the phenomena. The following section of this paper uses cross correlation of cumulative arrivals measured at successive detector stations. Then the paper discusses the limitations arising from flow-based measures, such as cumulative arrivals, when attempting to measure waves propagating against the flow of traffic. The next section uses traffic speed measurements to address the limitations and then examines the influence of prevailing traffic speed, lane, and location on the results. This paper then closes with a brief discussion and conclusions.
' Throughout this paper, speed refers to motion of the traffic as it travels downstream and velocity refers to the motion of waves through the traffic stream. Furthermore, a positive velocity is used to denote when these waves travel upstream.
Average velocity of waves propagating through congestedfreeway traffic 167
AVERAGE WAVE VELOCITY FROM CROSS CORRELATION OF CUMULATIVE ARRIVALS BETWEEN TWO STATIONS Our first attempt to extract wave velocities automatically is to simply replicate Munoz and Daganzo's cross correlation method on one month (August, 2003) of cumulative arrival curves sampled every 2-seconds from seven successive detector stations in the westbound direction of the Berkeley Highway Laboratory (BHL) along Interstate-80, north of Oakland, CA (Coifman et. al., 2000). The method seeks to find the time lag at which the cross correlation of the two time series recorded at two successive detector stations reaches a maximum and that time lag corresponds to the dominant travel time of waves in the sample. The details for the cross correlation method are as follows. Assuming that waves propagate between the two stations, the upstream time series, y(t), could be regarded as a scaled and delayed version of downstream time series, x(t), with zero mean additive noise, n(t), i.e., y(t) = ax(t-ro)
+ n(t)
(1)
where cris the scale factor and z0 is time delay corresponding to the travel time of waves from downstream location to upstream location. Let R (T) represent the cross correlation of x(t) and y(t) , and R^ij)
be the auto-correlation of x(t) . From equation (1) and the
definitions of Rxy{t) and Rxx(t), we have: Rxy(r) = E[xQ)y(t + r)] = E[x(t){ax(t + T-T0) + n(t + T)}] = aE[x(t)x(t + T-T0)] = ORXX(T-T0)
Q)
where Ra (r) < R^ (0). Thus, Rxy (r) reaches its maximum when r = r 0 . For this study the search window for feasible time lags spanned -4096 sec to 4096 sec and in the rare cases that two maxima were found, the given sample is excluded. There are 5 lanes in each direction on the freeway segment used in this study (fig. 1). The innermost lane is reserved for high occupancy vehicles (HOVs), because the HOV lane generally exhibits little congestion compared to the general flow lanes it was excluded from the analysis of upstream moving waves. The seven dual-loop detector stations in the segment are denoted stl through st7, respectively. The stations are spaced about 1/2 km apart from each other and each detector station has a dual-loop detector in each lane, which allows for direct measurement of local traffic speeds.
168
Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
The event data from each detector are recorded at 60 Hz and then aggregated to 2-second average traffic speed and flow measurements. This 2-second aggregation was chosen because it is roughly on the order of typical headways and is consistent with Munoz and Daganzo (2001), facilitating comparison. Unlike Munoz and Daganzo, who paired lanes together, we applied the cross correlation technique on individual lanes. Because this study is limited to the waves that arise in congested traffic, the following pre-processing step is used to exclude all non-congested periods. The lane is considered congested if the 5-minute average traffic speed remains below a pre-specified threshold for at least one-hour at both the upstream and downstream stations on a link. Each contiguous "congested period" in each lane for each station pair is divided into non-overlapping, one-hour long samples that are used in the cross correlation analysis. The exact threshold was chosen somewhat arbitrarily, with the specific intent to ensure that conditions were indeed congested throughout the entire link for an extended period. A traffic speed threshold of 48 km/h (30 mph) was used in most of the analysis presented in this paper, though a higher threshold at 72 km/h (45 mph) was used as well throughout the research to verify the results did not depend on the choice of the threshold. Other sample durations were studied, such as 45-min and 30-min long, with results similar to those presented herein, but they became noisier as the period decreased. Fig. 2 shows the cumulative distribution (CDF) of the average wave velocity from each sample, estimated via cross correlation on an individual lane basis applied to the 2-second cumulative arrival curves after subtracting a background flow equal to each lane's average flow during the given period. So each datum in this distribution represents the wave velocity corresponding to the maximum correlation from a one hour long sample for the given lane. Rather than reporting mean or median, the entire distribution is shown to allow detailed evaluation, e.g., the standard deviation could be evaluated from the CDF. Over all links and in all lanes, one sees mixed results. Many of the observations are consistent with the earlier research, with wave velocities between 15 km/h and 25 km/h, but many more of the measurements fall outside this range. For reference, table 1 shows the total number of samples in each of the curves from fig. 2. The research also considered several related variations. In the first case, before employing cross correlation, rather than subtracting a background flow the cumulative arrival curves were filtered to remove the least-squares fit of a straight line from the data. Next, the time series of 2-second "raw" flow and of "filtered" flow (using the same filtering processing as used in cumulative arrival curves) were investigated for
Figure 1.
Schematic of the Berkeley Highway Laboratory on Interstate-80, north of Oakland, California.
Average velocity of waves propagating through congested freeway traffic
169
estimating the average wave velocity, applied to the same congested periods used in fig. 2. Fig. 3 shows the resulting CDFs of each sample's average wave velocity from the three additional methods for a typical station pair (st4 and st5 westbound, the other station pairs exhibited similar trends), compare to fig. 2D. The filtered cumulative arrivals found almost half of the measurements with zero time lag because in many cases the background flow was the dominant component. As a result, fig. 3A shows the CDF with the zero time lag included as dashed lines and excluding all samples with zero time lag with solid lines for the same lanes. In any event, all four methods had more measurements between 15 km/h and 25 km/h than any other 10 km/h window, with the filtered flow having many more measurements in this range compared to the other three methods. Although encouraging, the results remain mixed for all four methods since several lanes had fewer than 40 percent of the observations falling in the window.
Figure 2.
Distribution of the average wave velocity from each sample using cross correlation applied to cumulative arrival curves with background flow subtraction, August 2003, WB, BHL.
170 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
Figure 3. Distribution of the average wave velocity from each sample using cross correlation applied to (A) filtered cumulative arrival curves (B) raw flow (C) filtered flow, st4 to st 5, August 2003, WB, BHL.
Table 1.
Number of Samples m each CDF for fig. 2 and fig. 5 Number of samples by lane st1 to st2
st2 to st3
st3 to st4
st4 to st5
st5 to st6
st6 to st7
Iane2
34
68
74
87
95
73
Iane3
62
70
84
75
76
63
Iane4
61
62
63
46
41
32
Iane5
44
44
56
37
13
10
PROBLEMS USING FLOW, OCCUPANCY AND CUMULATIVE ARRIVALS TO EXTRACT AVERAGE WAVE VELOCITY Given homogeneous vehicles and drivers, LWR predicts that for a convex flow-density relationship all waves should propagate upstream during congestion. Yet there are numerous examples in the literature demonstrating that some information actually travels downstream during congested periods, as observed in flow or occupancy measurements, e.g., Dailey (1993), Petty et al. (1997) and Cassidy and Windover (1998). Cassidy and Windover referred to this phenomenon as "driver memory" with drivers resuming similar headways after passing through disturbances while the other researchers simply used the phenomenon to measure travel time. None of these papers note the seemingly apparent conflict with LWR. To rectify this problem, one can derive an explanation based on the fact that real vehicles simply are not homogeneous.
Average velocity of waves propagating through congestedfreeway traffic 171 Consider fig. 4, showing a hypothetical example of several vehicles traveling in a freeway lane. The figure shows four vehicles moving from right to left. The i-th vehicle has a fixed effective length2, Lt, and a variable spatial gap between itself and the preceding vehicle, git that is likely to be a function of the driver, vehicles, and traffic conditions. The total spatial headway for this vehicle at any instant is simply, s
i = Li + Si
(3)
If two successive vehicles, i-1 and i, were traveling at the same constant speed, U, then the ith vehicle's headway passing a point in space is given by, h,=^-
(4)
or relaxing the assumption of constant speed to reflect the varying speeds experienced in congested traffic (as is done throughout the remainder of this paper), the relationship becomes, A,*^
(5)
where ut is the i-th vehicle's speed passing the specified point in space, e.g., a detector station. Similarly, as the vehicle passes a detector, it turns on for the following duration,
Figure 4.
2
A hypothetical example of the traffic stream showing the corresponding spacings, headways and detector status.
The effective length includes the physical length of the vehicle and the length of the detection zone.
172 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction L.
(6)
where an equal sign is used assuming the vehicle is traveling fast enough so that acceleration is negligible in the on-time measurement. Given these facts, if one measures flow, qJy and occupancy, ocCj, over a fixed number of vehicles, n, it should be clear that, n
1j=-
.
°CCj=—n
(7)
Most operating agencies use fixed sample periods, T, rather than fixed n, in which case the equalities in Equation 7 become "approximately equal" due to edge effects from partial headways observed at the start and end of a given sample.3 It should be clear from this derivation that both flow and occupancy are functions of ut and Lt. Obviously, Li travels downstream with the vehicles even during congested periods, e.g., Coifrnan and Cassidy (2002) explicitly used measured vehicle lengths to match the observation of a given platoon of vehicles at one station with the earlier observation of the same platoon from an upstream station on the same freeway during congested periods. This information traveling downstream will confound any attempt to extract information propagating upstream against the flow of traffic, e.g., although cut off by the horizontal axis of the plots, many velocities corresponding to downstream moving waves were found in the CDFs of fig. 2. The velocity of many of these downstream moving waves were on the order of the prevailing traffic speed, but they were not the dominant trend in the distribution, as evidenced by the majority of the velocities in fig. 2 indicating upstream moving waves. The relatively low frequency of downstream moving waves in the CDFs is due in part to the large sample period (1 hr), during which time the link travel speed experienced by drivers typically changes significantly and this range usually prevents the downstream moving waves from becoming the dominant feature in the crosscorrelation. The existence of downstream moving waves during congestion is consistent with the theoretical development of the mixed flow first-order macroscopic models of Zhang and Jin (2002) and Chanut and Buisson (2003), which extend LWR to heterogeneous flows of two or more vehicle types. Namely, in their models, trucks and cars had different lengths but travel with the same speed during congestion. These models predict that in congestion there are two kinds of waves, waves that move downstream with the vehicles as an artefact of the inhomogeneous vehicle (and driver) fleet and waves that move upstream according to LWR. If ui is independent of the inhomogeneities of the vehicle fleet, then one would expect that during congested periods LWR would better predict the evolution of traffic speed over time
3
In this case n changes from one sample to the next while T is the summation of the headways during a given sample. Provided the sample includes many vehicles, the contributions from the partial headways at the start and the end of the sample are usually negligible.
Average velocity of waves propagating through congestedfreeway traffic 173 and space than LWR does for flow during the same periods (or cumulative arrivals, which is simply a function of flow).
AVERAGE WAVE VELOCITY FROM CROSS CORRELATION OF LOCAL TRAFFIC SPEED BETWEEN TWO STATIONS In an attempt to improve the performance of the automated data extraction, and thus to analyze the characteristics of waves, the analysis now shifts to using local traffic speed from the detector stations to measure the average wave velocity in each hour long sample. As with the cumulative arrival data, the time series of local traffic speed measurements were filtered to remove the least-squares fit of a straight line from the data. Fig. 5 shows the CDFs of average wave velocity from each sample, estimated using the time series of 2-second traffic speed measurements corresponding to the same congestion samples used in fig. 2. The CDFs from local traffic speed exhibit a much tighter distribution compared to those from the cumulative arrival curves or flow measurements. Now all lanes in all links have over 90 percent of the measured wave velocities between 15 km/h and 25 km/h. Grouping all of the lanes and station pairs together, the median wave velocity is 18 km/h. Note that fig. 2 and fig. 5 only show the results for westbound traffic in August 2003 (WB03). The analysis was repeated for eastbound traffic (EB-03), as well as data from August 2002 (WB-02 and EB-02). In all four cases the results exhibited the same pattern moving from cumulative arrivals to local traffic speed when measuring the average wave velocity during congestion and all four cases exhibited the dominant wave velocity response in the range of 15-25 km/h, which is consistent with the previous research.
Characteristics of waves To investigate whether the upstream moving wave velocity is independent of local traffic conditions, this research examined the relationship between average wave velocity and the prevailing traffic speed in each sample. For each one-hour long congestion sample we measured the minimum, median, and maximum of the 2-second traffic speed measurements from the downstream detector station. The results were similar for all three parameters, so this paper only presents the results for the median. The points in fig. 6 show each sample's average wave velocity versus the median of the downstream station's traffic speed from the same data set presented in fig. 5. For brevity, all lane and station pairs are shown in a single plot. Of the 1370 data points in the figure only 8 percent are below 15 km/h. To extract the general trend, these data were binned by 5 km/h increments of the local traffic speed and the total number in each bin is noted in the first column of table 2. The median of each bin is shown in fig. 6 as a solid line, with the median being chosen over the mean to reduce the sensitivity to the small number of points that are far from the center of the distribution. Note that the slowest bin had
174 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 2. prevailing traffic speed range 0-5 km/h 5-10 km/h 10-15 km/h 15-20 km/h
Number of samples in fig. 6 - 9 Subsampling WB-02 WB-03 0
EB-03 0
5 71
3
WB-02 0
EB-02 0
0
0
0
0
2
34
0 3 21
3 12
20-25 km/h
184 271
15
71 104
25-30 km/h 30-35 km/h
310 286
48 61
132 111
55 82
35-40 km/h
197
40-45 km/h 45-50 km/h
45 1
92 26
66 13
4
0
Figure 5.
onramps 0
offramps lane 2 lane 3 lane 4 lane 5 0 0 0 0 0 0 1 2 2 2 18 31 30
8 49
27
113
63 85
33
107
96
18 24
91
3
55 8
101 52
20
43 28 2 0
0
0
0
0
84
18
5 0
21 42
15 30
49
24
58 64
49 30
51 17 1
39 15 0
Distribution of the average wave velocity from each sample using cross correlation applied to filtered traffic speed measurements, August 2003, WB, BHL.
only 5 samples and the fastest bin had only 1 sample. Ignoring the larger deviations arising from these two bins on the ends, the fitted line is relatively flat, ranging between 17 km/h and 19 km/h. Fig. 7 presents all four of the fitted lines resulting from each direction in each year.
Average velocity of waves propagating through congested freeway traffic
175
As with fig. 6, table 2 tallies the number in each bin, and again, both the first and last bin for each curve has few samples, they are included only for completeness. For each curve in fig. 7, one can see the wave velocities may exhibit a small correlation with the prevailing traffic speed, but given the fact that three of the curves increase slightly and one decreases, it would be premature to draw any conclusions about the details of any relationship. Investigating these finer points of the relationship will be the subject of future research. In any event, the range is less than 4 km/h of wave velocity over a span of 35 km/h of local traffic speed for all four of the curves. So if local traffic speed impacts the wave velocity during congested periods this impact is small and there are other unmeasured factors that impact the wave velocity with a similar magnitude. The congestion threshold was set to 48 km/h in fig. 7, but as noted previously, the analysis was repeated with a threshold of 72 km/h. The results from the higher threshold generally exhibit the same pattern but become slightly noisier at higher traffic speeds because many of the one-hour samples tend to include a mixture of free flow and congested conditions.
Figure 6.
Average wave velocity versus prevailing traffic speed for each sample, August 2003, WB. BHL.
176 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
Figure 9.
(A) Median wave velocity versus prevailing traffic speed by lane, August 2003, WB, BHL. (B) Detail of A
Finally, we explicitly investigate if there is a strong relationship between lane change maneuvers and wave velocity. Because it is difficult to measure lane change maneuvers directly, inflow is used as a proxy for the net number of lane change maneuvers during a given sample, where the inflow is defined as the cumulative downstream arrivals less the cumulative upstream arrivals in the lane over the hour. Inflow will underestimate the total number of lane change maneuvers since exiting vehicles cancel out entering vehicles during the sample. Noise also arises in the measurement due to the fact that the number of vehicles stored in the lane may change between the start and the end of the sample, though this noise should be unbiased. Fig. 10 shows inflow versus wave velocity. Lane 5 regularly exhibits an outflow (negative inflow) of 500 vph between st 1 and 2, and then an inflow of 400 vph between st 2 and 3 due to the ramps. A few of the other lanes have an inflow on the order of 200 vph for these links, with the magnitude of the inflow being much smaller elsewhere. Yet as shown in Fig. 5 for these same data, all four lanes exhibit similar trends in wave velocity over all of the links. Thereby suggesting that aggregate wave velocity is independent of inflow. Of course it is possible that a relationship to lane change maneuvers may have been obscured by the long sampling period, the use of a proxy measurement, or simply the noise in the data.
CONCLUSIONS This paper employed cross correlation to estimate wave velocities between several successive detector stations during congested periods and over a large data set. Given homogeneous vehicles and drivers, LWR predicts that for a convex flow-density relationship all waves should propagate upstream during congested periods. The analysis first employed cumulative arrivals — a function of flow — and then the local traffic speed at the detector stations. It was shown that the flow-based analysis yield mixed results, with many measurements being consistent with earlier research, but many more measurements falling outside the typical range of measured wave velocities from the literature. But vehicles are not homogeneous and it was also shown that flow and occupancy depend on effective vehicle length as well as the
Average velocity of waves propagating through congestedfreeway traffic 177
Figure 10. Inflow versus wave velocity from each sample using cross correlation applied to filtered traffic speed measurements, August 2003, WB, BHL. local traffic speed. Because the vehicle lengths travel downstream with the vehicles, this information will hinder attempts to extract wave velocities propagating against the flow of traffic when using flow-based measures. This fact is evident in fig. 2 and 3. Trucks are restricted from lane 2, thus the standard deviation of vehicle lengths is smaller and less information travels with the vehicles. As a result, comparing across lanes for all of the plots in these figures, lane 2 had the highest percentage of observations that fell within the 15 km/h to 25 km/h window. To reduce the influence of the confounding vehicle lengths, the analysis was repeated over the exact same samples using local traffic speed measurements. Because drivers are constrained by downstream vehicles in congestion, the traffic speed trends are much less dependent on specific vehicle or driver characteristics. It was confirmed that LWR does a better job predicting the evolution of average traffic speed over time and space, with over 90 percent of the samples having an average wave velocity propagating upstream in the range of 15 km/h to 25 km/h. Of course these results represent the aggregate performance over one-hour samples and individual waves could differ significantly from this range, e.g., it is likely that other sources of noise remain in the data, such as lane change maneuvers disrupting the propagation ofwaves. Unlike earlier studies, the method presented in this paper is not labor intensive, thereby allowing for the study of a large amount of data and extracting the general trends in the wave velocity during congestion. Our results over two months of data from both westbound and
178 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction eastbound lanes of the BHL show that wave velocities are nearly constant over a large range of local traffic conditions, which is consistent with earlier work and further supports the use of a triangular flow-density relationship at least for a first-order approximation. There was some evidence that the wave velocity may change with local traffic speed but the change is very small, less than 1 kmyh of wave velocity for every 10 km/h of local traffic speed. Given the fact that the direction of this trend changed across the four data sets, this small trend may simply be an artifact of the sample size or it is indicative of other factors that have a similar impact on the wave velocity but are yet to be identified, hi any event, these outstanding questions are the subject of ongoing research. Unlike most of the earlier efforts to measure wave velocity during congested periods, the analysis was applied on a lane-by-lane basis, allowing for comparisons between lanes and for the analysis to quantify the impact on the wave velocity from vehicles entering or leaving at ramps. To this end, wave velocity does not appear to have a dependence on ramps or the particular lane. However, future research will seek to verify this finding. For example, this analysis was limited to the BHL, which has five lanes in each direction, results may be different for a two or three lane freeway.
ACKNOWLEDGEMENTS This material is based upon work supported in part by the National Science Foundation under Grant No. 0133278. The authors also wish to thank the anonymous reviewers for their valuable feedback.
REFERENCES Cassidy, MJ. and J.R. Windover (1998). Driver memory: motorist selection and retention of individualized headways in highway traffic. Transportation Research: Part A, 32(2), 129-137. Chanut, S. and C. Buisson (2003). Macroscopic Model and Its Numerical Solution for TwoFlow Mixed Traffic with Different Speeds and Lengths. Transportation Research Record: Journal of the Transportation Research Board, No. 1852, TRB, National Research Council, Washington, D.C., pp.209-219. Coifman, B., D. Lyddy and A. Sabardonis (2000). The Berkeley Highway LaboratoryBuilding on the 1-880 Field Experiment. Proc. IEEE ITS Council Annual Meeting, IEEE, 2000, pp 5-10. Coifman, B. (2002). Estimating Travel Time and Vehicle Trajectories on Freeways Using Dual Loop Detectors. Transportation Research: Part A, 36(4), pp. 351-364. Coifman, B., and MJ. Cassidy, (2002). Vehicle Reidentification and Travel Time Measurement on Congested Freeways, Transportation Research: Part A, 36(10), pp. 899-917. Daganzo, C. F. (1997). Fundamentals of Transportation and Traffic Operations. Pergamon.
Average velocity of waves propagating through congestedfreeway traffic 179 Dailey, D. (1993). Travel Time Estimation Using Cross Correlation Techniques, Transportation Research: Part B, 27(2), pp 97-107. Hall, F. L., B. L. Allen and M. A. Gunter (1986). Empirical Analysis of Freeway FlowDensity Relationships. Transportation Research: Part A, 20(3), pp. 197-210. Hall, F. L and T. N. Lam (1988). The Characteristics of Congested Flow on a Freeway across Lanes, Space and Time. Transportation Research: Part A, 22(1), pp. 45-56. Lighthill, M. J. and G. B. Whitham (1955). On Kinematic Waves. I Flow Movement in Long Rives. II A Theory of Traffic Flow on long Crowded Roads. Proc. Roy. Soc, A. 229, 281-345 Munoz, J. C. and C. F. Daganzo (2001). The Bottleneck Mechanism of a Freeway Diverge. Paper presented at the 80th annual TRB meeting, Transportation Research Board, January 2001. Mauch, M. and M. J. Cassidy (2002). Freeway traffic oscillations: observations and predictions. International Symptom of Traffic and Transportation Theory, (M.A.P. Taylor, Ed.) Elsevier, Amsterdam, pp. 653-674. Newell, G. F. (1993). A Simplified Theory of Kinematic Waves in Highway Traffic, I General Theory, II Queuing at freeway bottlenecks, III Multi-destination Flows. Transportation Research, Part B, 27(4), pp. 281-313. Petty, K., P. Bickel, M. Ostland, J. Rice, F. Schoenberg, J. Jiang, and Y. Ritov, (1997). Accurate Estimation of Travel Times From Single Loop Detectors, Transportation Research: Part A, 32(1), pp 1-17. Richards, P. I. (1956). Shockwaves on the Highway. Operations Research, pp. 42-45. Smilowitz, K. R. and C. F. Daganzo (2002). Reproducible Features of Congested Highway Traffic. Mathematical and Computer Modeling, 35, 509-515. Windover, J. and M. J. Cassidy (2001). Some Observed Details of Freeway Traffic Evolution. Transportation Research Part A, 35(10), pp. 881-894. Zhang, H.M. and W.L. Jin (2002). Kinematic Wave Traffic Flow Model for Mixed Traffic. Transportation Research Record: Journal of the Transportation Research Board, No. 1802, TRB, National Research Council, Washington, D.C., pp. 197-204.
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10
MICROSCOPIC THREE-PHASE TRAFFIC THEORY AND ITS APPLICATIONS FOR FREEWAY TRAFFIC CONTROL Boris S. Kerner, DaimlerChrysler AG, Stuttgart, Germany
INTRODUCTION There are a huge number of traffic flow theories and models, which should explain diverse congested traffic patterns that are observed upstream of freeway bottlenecks (see e.g. references in the reviews by Gartner et al. (eds.) (1997); Helbing (2001)). However, it is only recently that a "puzzle" of spatiotemporal features of congested patterns has been solved and these pattern features adequately understood (Kerner, 1998, 2002). Consequently, earlier traffic flow theories and models are in a serious conflict with many of these empirical spatiotemporal traffic pattern features. Therefore, the author introduced "three-phase traffic theory" (see references in Kerner (2004)). In three-phase traffic theory, besides the "free flow" phase there are two other phases in congested traffic: "synchronized flow" and "wide moving jam." Thus, there are three traffic phases in this theory: 1. Free flow. 2. Synchronized flow. 3. Wide moving jam. Three-phase traffic theory explains the complexity of traffic based on phase transitions among the three traffic phases, and on their complex nonlinear spatiotemporal features. The empirical (objective) criteria that distinguish between the two traffic phases in congested traffic are related to spatiotemporal features of these phases (Kerner, 1998, 2002). A wide moving jam is a moving jam that maintains the mean velocity of the downstream jam front, even when the jam propagates through any other traffic states or freeway bottlenecks. In
182 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction contrast, the downstream front of the "synchronized flow" phase is often fixed at a freeway bottleneck. Within this front, vehicles accelerate from lower speeds in synchronized flow to higher speeds in free flow. Three-phase traffic theory is a qualitative theory (Kerner, 1998, 2002). The related microscopic theory (Kerner and Klenov, 2002, 2003) confirms hypotheses of this theory. This microscopic theory explains all known up to now reproducible empirical spatiotemporal congested pattern features at freeway bottlenecks. Note that in the book (Kerner, 2004) empirical features of reproducible freeway traffic patterns, three-phase traffic theory including a microscopic analysis of congested patterns, as well as their engineering applications (traffic pattern recognition, tracking, prediction, and control) have been considered. In this paper, simulation model results based on a numerical study of a microscopic traffic flow model in the context of three-phase traffic theory are presented. A deep connection between the diagram of congested patterns at bottlenecks with probabilistic theory of freeway capacity of free flow at bottlenecks, as well as application of the microscopic three-phase traffic theory for on-ramp inflow control and for a study of an ACC vehicle influence on spatiotemporal congested patterns at bottlenecks are discussed. STOCHASTIC MICROSCOPIC MODEL Here, we consider a stochastic microscopic two-lane model in the context of three-phase traffic theory (Kerner and Klenov, 2002, 2003). Model results presented below are related to numerical simulations of this model. We discuss only a hypothetical traffic flow with identical vehicles. A model of heterogeneous flow can be found in (Kerner, 2004). Basic rules of vehicle motion Basic rules of vehicle motion in the model of identical vehicles are as follows: vn+1 =max(0,min(v f r e e ,v s n ,v c n )),
Vc
jVn+An - "|vn+ant = n
f
Or for
x n+1 = x n + vn+1x,
«,n-Xn^Dn x,,n-xn>Dn'
(1)
X
An = max(-b n x,min(a n x,v M -v n )),
U
(3)
where index n corresponds to the discrete time t = nx, n = 0,1,2,...; x is the time step; vfrl,e is the maximum speed in free flow; the lower index £ marks functions (or values) related to the preceding vehicle; D n is a synchronized distance; an > 0 and b n > 0 (see below). In (1), vs n is a safe speed (KrauB et ah, 1997) that is a solution of the equation (Gipps, 1981)
Microscopic three-phase traffic theory and applications 183 v s ,^ + X d (v Sjn ) = g n +X d (v S j n ),
whereX d (u) = bT2(ap + a ( a - l ) / 2 ) ,
(4)
g n = x ^ n - x n - d is the space gap between vehicles, d is the vehicle length, b is a constant, a is the integer part of u / bx, (3 is the fractional part of u / bx. Speed adaptation within synchronized distance and steady model states Eqs. (l)-(3) describe the speed adaptation effect in synchronized flow. This speed adaptation effect takes place when the vehicle cannot pass the preceding vehicle, within the synchronization distance D n : at xn - x n n ),
(5)
D(u, w) = d + max(0, ktu + (^"'u^i - w)),
(6)
where k>l and (j> are constants; a is the maximum acceleration. Model fluctuations and random vehicle acceleration and deceleration As in other stochastic traffic flow models (see references in Helbing (2001)), at the first step of calculations of the model, the speed of each vehicle is vn+1 = v n+1 ,
(7)
where vn+1 is calculated based on Eqs. (l)-(6). At the second step, a noise component ^n is added to the calculated speed vn+1 (7) and then the final value of the speed vn+1 at time step n+1 is found from the condition vn+1 = max(0, min(vfree, vn+1 + ^ n , vn + a, v s n )).
(8)
Random deceleration and acceleration are applied depending on whether the vehicle decelerates or accelerates, or else maintains its speed: f-5b 4n =
if Sn+1 = -1
4a i f S n + 1 = l { 0 ifS n + 1 =O,
(9)
184 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction where 4 a and t,h are random sources for deceleration and acceleration, respectively: (10) (11)
^=aT6(pa-r), *b=aT0(Pb-r),
p a and p b are probabilities of random acceleration and deceleration, respectively; r = rand(0,l), i.e., this is an independent random value uniformly distributed between 0 and 1; 9(z) = 0 at z < 0 and 9(z) = 1 at z > 0; S n+] denotes the state of vehicle motion (S n+1 = - 1 represents deceleration, Sn+1 = 1 acceleration, and Sn+1 = 0 motion at nearly constant speed):
Sn+1=
[-1 i f v n + 1 < v n - 8 1 ifvn+1>vn+8 0 otherwise,
(12)
where 8 is a constant (8 « at).
Fig. 1. Explanation of the model: (a) Hypothetical steady state model solutions, (b) Steady state solutions and the line / that represents a steady propagation of the downstream front of a wide moving jam. (c) Model of an on-ramp bottleneck. In (b), qout and p min are the flow rate and density in free flow formed in the wide moving jam outflow, p max is the density within a wide moving jam (see references in Kerner (2004)). To simulate a driver time delay either in vehicle acceleration or in vehicle deceleration, an and b n in (2), (3) are taken as the following stochastic functions an=a6(P0-ri), bn=a0(P1-r1),
(13) (14)
Microscopic three-phase traffic theory and applications 185
Po=!Po 0 [1
if
*-*1 Sn=l,
Pl
lf
S
lP 2
if
Pl=j 1
lf
"""1
(15)
(16)
Sn=-1,
where rx = rand(0,1); probabilities p o (v), p 2 (v) are given functions of speed, probability P! is a constant;
1 - Po and 1 - P, are the probabilities of a random time delay in vehicle
acceleration and deceleration, respectively. Models for two-lane traffic flow and on-ramp bottleneck Lane changing rules in a two-lane model are associated with conditions R^L:
v;>v,iI,+81andvI1Svn,
(17)
L^R:
v;>v,>n+51orv>vn+51
(18)
for lane changing from the right lane to the left (passing) lane (R -» L) (17) and a return change (L —» R) (18). The safety conditions for lane changing are g>min(vnT,D;-d),
(19)
g;>min(v;T,D;-d),
(20)
D;=D(vnX),
(21)
D;=D(V;,vn).
Functions in (21) are associated with the function D(u,w) (6). In (17), (18), &l >0 is a constant, hi (17)-(21), superscripts + and - in variables and functions denote the preceding vehicle and the trailing vehicle in the "target" (neighboring) lane, respectively. In the model, there is an on-ramp on the main two-lane road (Fig. l(c)). The related on-ramp bottleneck consists of two parts: (i) The merging region of length L m , where vehicles can merge onto the main road from the on-ramp lane, (ii) A part of the on-ramp lane of length Lr upstream of the merging region, where vehicles move in accordance with the model (1)-(16). Open boundary conditions are applied. Main behavioral model assumptions and model parameters 1. In synchronized flow, a driver accepts a range of different hypothetical steady state speeds at the same space gap to the preceding vehicle (see (2) and Fig. l(a)). 2. A driver tends to adjust the speed to the preceding vehicle within the synchronization distance (5), (6).
186 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 3. Over-acceleration effect: In synchronized flow of a lower density, a driver searches for the opportunity to accelerate and to pass. Random acceleration ^a (9), (10) describes this effect: The over-acceleration is simulated as a "collective effect", which occurs on average in traffic flow, through the use of random vehicle acceleration. 4. Over-deceleration effect occurring if the preceding vehicle begins to decelerate unexpectedly. This well-known effect associated with a driver time delay in deceleration simulates moving jam emergence in synchronized flow. In the model, the over-deceleration is simulated as a "collective effect" through the use of random deceleration £b (9), (11). Thus, random deceleration and acceleration (9)-(ll) simulate driver time delays mentioned in item 3 and 4, which occur if the vehicle either accelerates or decelerates only, therefore, in (9) £,n = 0 if Sn+1 = 0. 5. At the downstream front of synchronized flow, which separates this synchronized flow upstream and free flow downstream, a driver in synchronized flow does not accelerate before the preceding vehicle has begun to accelerate. Formulae (13), (15) together with (l)-(3) simulate the associated driver time delay in acceleration that takes place if the vehicle does not accelerate (S n * 1) at time step n. 6. Moving in synchronized flow, a driver comes closer to the preceding vehicle over time that explains the pinch effect in synchronized flow (Kerner, 2004). Formulae (14), (16) together with (l)-(3) simulate a driver time delay in deceleration associated with this effect. We see that random model characteristics are used to simulate various driver time delays, which are assumed to be different in various local driving conditions. Quantitative validation of these driver time delays and of other model parameters have been made to have simulation parameters of congested patterns, in particular, the characteristic times of phase transitions as well as the velocities of the fronts that separate free flow, synchronized flow, and moving jams within the patterns in accordance with empirical results (Kerner, 2002). A detailed consideration of this model parameter validation can be found in (Kerner, 2004). DIAGRAM OF CONGESTED PATTERNS AT ON-RAMP BOTTLENECK IN THREE-PHASE TRAFFIC THEORY This section contains simulation results of congested patterns at an on-ramp bottleneck and of the associated congested pattern diagram determined from application of the model presented in the previous section. This congested pattern diagram contains regions of congested pattern occurrence and existence in the flow-flow plane (Fig. 2(a)) whose coordinates are the flow rate qin in free flow upstream of the bottleneck and the flow rate to the on-ramp qon (Fig. l(c)) (Kerner, 2002; Kerner and Klenov, 2002, 2003). There are two main types of congested patterns (Fig. 2): (i) a synchronized flow pattern (SP) and (ii) a general pattern (GP). The SP is a congested pattern that consists of synchronized flow only, i.e., there are no wide moving jams within the SP. The GP is a congested pattern, which consists of synchronized flow upstream of the bottleneck and wide moving jams that emerge spontaneously in that synchronized flow.
Microscopic three-phase traffic theory and applications 187 There are three types of SPs: (1) Widening SP (WSP) (Fig. 2(b)). The downstream front of an WSP is fixed at the bottleneck. The upstream front of the WSP propagates upstream continuously over time. (2) Localized SP (LSP) (Fig. 2(c)). As in the WSP, the downstream front of an LSP is fixed at the bottleneck. In contrast with the WSP, the upstream front of the LSP is localized at some distance upstream of the bottleneck. (3) Moving SP (MSP) (Fig. 2(d)). In contrast with the LSP and WSP, an MSP is an SP that propagates as a whole localized pattern on the freeway over time. However, in contrast with a wide moving jam, if an MSP reaches a bottleneck, the MSP is caught at the bottleneck: The MSP that propagates upstream can exist only for a finite time. There are two main boundaries in the pattern diagram, FS(B) and Sj B) (Fig. 2(a)). Below and left of the boundary FgB), free flow is realized at the bottleneck. Between the boundaries FS(B) and Sj B) different SPs emerge on the main road upstream of the bottleneck (Figs. 2(b-d)). Right of the boundary Sf } different GPs appear (Figs. 2(e-h)). If qin > q out , where qout is the flow rate in free flow formed in the wide moving jam outflow, then the width (in the longitudinal direction) of the farthest upstream wide moving jam of an GP gradually increases over time (Figs. 2(e, h)). If in contrast qin < q out , then the farthest upstream wide moving jam of an GP gradually dissolves over time (Fig. 2(f)). However, even if qin < q out , the region of wide moving jams in the GP can expand in the upstream direction. There is a region in the diagram right of the boundary Sj B) where a dissolving GP (DGP) occurs. In the DGP (Fig. 2(g)), continuous wide moving jam emergence is interrupted. As a result, a single wide moving jam emerges only and an LSP remains at the bottleneck, while the wide moving jam moves on the main road upstream of the bottleneck. It must be noted that several independent scientific groups have recently developed new microscopic traffic flow models in the context of the author's three-phase traffic theory (Davis, 2004; Lee et at, 2004; Jiang and Wu, 2004). Models of Lee et al. (2004) and Jiang and Wu (2004) exhibit features of the diagram of Kerner (2002) (Fig. 2(a)).
188 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
Fig. 2. Congested patterns at on-ramp bottleneck (Kerner, 2002; Kemer and Klenov, 2002, 2003): (a) - Congested pattern diagram, (b-g) - Vehicle speed on the main road within different congested patterns: an WSP (b), an LSP (c), an MSP (d), an GP at qin > qout (e), an GP at qin < qout (f), and an DGP (g). (h) - Vehicle trajectories within the GP in (e).
Microscopic three-phase traffic theory and applications 189 CRITICAL DISCUSSION OF EARLIER DIAGRAM OF CONGESTED PATTERNS AT ON-RAMP BOTTLENECK In this section, we compare the congested pattern diagram of three-phase traffic theory discussed in the previous section with earlier model results associated with congested pattern emergence at an on-ramp bottleneck as well as with empirical features of the onset of congestion in free flow at the bottleneck. Results of three-phase traffic theory (Kerner, 1998, 2002, 2004) contradict qualitatively with conclusions of all earlier traffic flow theories and models that claim to explain the onset of congestion at the bottleneck (Helbing, 2001). Most of these microscopic, macroscopic, probabilistic, and other models of freeway traffic explain the onset of congestion in free flow at the bottleneck by moving jam emergence. Note that congested states occurring at the bottleneck, which are associated with different types of moving jams, are classified by Helbing et al. (1999) as MLC (moving localized cluster), TSG (triggered stop-and-go traffic), and OCT (oscillatory congested traffic) (Fig. 3(a)). (a) Diagram of Helbing
Fig. 3. Sketch of congested pattern diagrams at an on-ramp bottleneck at a high enough given flow rate qin upstream of the bottleneck as a function of the flow rate to the on-ramp qon : (a) - earlier traffic theories (Helbing et al, 1999). (b) - three-phase traffic theory (Kerner, 2002). However, the fundamental model result that the onset of congestion in free flow at freeway bottlenecks is associated with spontaneous moving jam emergence (Kerner et al, 1995; Lee et al, 1998; Helbing et al., 1999; Helbing, 2001) is in a serious conflict with empirical evidence (Kerner, 1998, 2002, 2004). In empirical observations, rather than moving jam emergence, a phase transition from free flow to synchronized flow (F -> S transition) governs the onset of congestion in free flow at the bottleneck. A first-order F —> S transition postulated in three-phase traffic theory discloses the nature of the breakdown phenomenon at freeway bottlenecks found in empirical observations (see Hall et al. (1992); Elefteriadou et al. (1995);
190 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Persaud et al. (1998); Kerner (2004) and references therein): The terms "F->S transition", "breakdown phenomenon", "traffic breakdown", and "speed breakdown" are synonyms. The crucial difference between the congested pattern diagram at an on-ramp bottleneck derived by Helbing et al. (1999) for earlier traffic flow models (e.g., Kerner et al, 1995; Lee et al., 1998; Helbing et al., 1999; Helbing, 2001) and the congested pattern diagram of threephase traffic theory (Fig. 2(a)) derived by Kerner (2002) can be seen in Fig. 3. If the flow rate upstream of the bottleneck qin is a given high enough value and the flow rate to the on-ramp qon increases gradually beginning from zero, then in the diagram of Helbing (Fig. 3(a)) the onset of congestion at the bottleneck is associated with moving jam emergence in free flow. This is inconsistent with empirical results mentioned above in which moving jams do not emerge spontaneously in free flow at the bottleneck. In contrast, in the diagram of Kerner (Fig. 3(b)) under the same conditions the onset of congestion is associated with synchronized flow emergence (different SPs appear). This is in accordance with empirical results. If qon further increases, then as follows from the diagram of Helbing (Fig. 3 (a)), homogeneous congested traffic (HCT) occurs in which no wide moving jams emerge. This model result is also inconsistent with empirical results in which at greater qon moving jams emerge in synchronized flow, i.e., GPs appear (Kerner, 1998, 2002). The latter is exact the result of the diagram of Kerner (Fig. 3(b)) in which GPs emerge at greater qon . Conclusions: Congested patterns at the bottleneck are qualitatively different in the diagram of Helbing (Fig. 3(a)) and in the diagram of Kerner (Fig. 3(b)). The diagram of Helbing is inconsistent with empirical congested pattern emergence, whereas the diagram of Kerner explains empirical congested pattern features. PROBABILISTIC THEORY OF FREEWAY CAPACITY AT BOTTLENECKS In this section, a deep connection between the congested pattern diagram based on threephase traffic theory (Figs. 2(a) and 4(a)) with a probabilistic theory of freeway capacity at bottlenecks is discussed (see references in Kerner (2004)). This diagram (Fig. 4(a)) found in model simulations and a probabilistic theory of traffic breakdown at an on-ramp bottleneck (Kerner and Klenov, 2005) can explain empirical features of probabilistic freeway capacity at the bottleneck (Elefteriadou et al., 1995; Persaud et al, 1998; Lorenz and Elefteriadou, 2000). Firstly note that in three-phase traffic theory (Kerner, 2004), traffic breakdown is associated with a deterministic, i.e., motionless and permanent perturbation (decrease in speed and increase in density) in free flow occurring at the bottleneck due to the on-ramp inflow. Even if there were no random perturbations in traffic flow, the deterministic traffic breakdown (deterministic F —> S transition) occurs within the deterministic perturbation at the bottleneck after a critical flow rate for deterministic traffic breakdown Isum
=
Ideterm, FS
(22)
Microscopic three-phase traffic theory and applications 191 is reached, where qsum = qon + q in . However, real random perturbations within the deterministic perturbation can cause random traffic breakdown (random F -> S transition) at the flow rate Isum
S transition is less than 1. The more distant the point (q on , q in ) from the boundary FS(B), the less the probability P^B) for an transition at the bottleneck. There should be the threshold boundary FtbB) in the diagram: If a
192 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction point (q o n ,q i n ) in the diagram is below and left of this threshold boundary Ft(hB), then the probability PpB' for an F -» S transition at the bottleneck during the time interval Tob is equal to zero, i.e., no F —> S transition can occur at the bottleneck during this time interval. Thus, there are an infinite number of maximum freeway capacities in free flow at an on-ramp bottleneck, qJ^exeB). These maximum freeway capacities are equal to the critical (maximum) flow rates downstream of the bottleneck qsiim = qin + qon associated with the points (q on , q in ) in the diagram, which lie on the boundary FS(B): qSx eB) =qsu m (qon.q m )l F . B .The maximum freeway capacities related to the boundary FS(B)
(24) depend on the flow rates qon ,
q in , and the time interval T ob . The greatest of these maximum freeway capacities is the maximum freeway capacity at qon -> 0 (but qon * 0 ) denoted by q^raexeB^ (Figs. 2(a) and 4(a)). Each of the maximum freeway capacities cannot be observed at the bottleneck during a longer time interval thanTob . This is related to an F—»S transition that occurs spontaneously in this free flow at the bottleneck during the time interval Tob with the probability
p B>
- L,>r> =1 -
(25)
If Tob decreases, then the position of the boundary FS(B) changes: At a given q in , the maximum capacity q™aexeB) associated with a new boundary position increases. This increase has a limit given by the maximum capacity q ^ ^ p s (22) that lies on a limit boundary Fdeterm s
m tne
diagram associated with a deterministic F —> S transition (deterministic traffic
breakdown) (Fig. 4(b)). Thus, when q££ B) -> q FS , the boundary F Fd(B)erm, s . There are also an infinite number of minimum freeway capacities at the bottleneck, q(tB). These minimum freeway capacities are equal to the threshold flow rates downstream of the bottleneck qsum = qin + qon associated with the points (q on , q in ) in the diagram, which lie on the threshold boundary Ft(hB) in the diagram of congested patterns: q!hB)=qsum(qon>q,n)|Fr-
(26)
The minimum freeway capacities are the minimum flow rates downstream of the bottleneck at which an F—>S transition can still occur spontaneously in this free flow at the bottleneck during the time interval T ob . This mean that the flow rates downstream of the bottleneck qsum = qin +qn m at are related to points (q on , q in ) in the diagram, which lie below and left
Microscopic three-phase traffic theory and applications 193 of the threshold boundary Ft(hB) in the diagram of congested patterns, are less than any freeway capacity: At these flow rates qsum = qin + qon the probability PFB) for an F —> S transition at the bottleneck during the time interval Tob is equal to zero. Maximum freeway capacities are related to the boundary FS(B) in the diagram. Minimum freeway capacities are related to the threshold boundary FtbB) in the diagram (Fig. 4(a)). Different flow rates qsum = qin + qon downstream of the bottleneck associated with the points (q on , q in ) between the boundaries FgB) and Ft(hB> in the diagram are also equal to the infinity of freeway capacities in free flow at the bottleneck. This is because at each of the corresponding flow rates qsum =q i n +q on an F—>S transition occurs spontaneously at the bottleneck during the time interval Tob with the probability P F B) , which satisfies the condition O 0. The equality of the
in Figs. 6(a, b)
is realized due to higher flow rate qin in free flow upstream
of the upstream bottleneck in Fig. 6 (a) in comparison with the flow rate qin in Fig. 6(b). This is also valid for Figs. 6(c, d). Depending on the flow rate q|^p) and on the congested pattern type at the downstream bottleneck, different intensification phenomena for the initial downstream congestion are found. We consider only two of them.
Transformation of DGP into GP An initial DGP at the on-ramp 'D', where only one wide moving jam emerges (Fig. 6(a)), transforms into an GP at this bottleneck where an uninterrupted sequence of wide moving jams emerges (Fig. 6(b)). After the wide moving jam in the DGP (Fig. 6(a)) emerges, the
196 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction current flow rate upstream of the on-ramp 'D' q,-nOwn:> cannot be greater than qout (Fig. l(b)): qtdown) qoni)- hi ANCONA, there is no on-ramp control as long as free flow is at the bottleneck. On-ramp inflow control is first realized only after the onset of congestion has occurred at the bottleneck. Feedback control is performed when v(det) < v cong , i.e., the average speed v(det) is equal to or drops below v cong . Then the flow rate qon°nt) is reduced via light signal operation. The decrease in this flow rate should lead to an increase in v(det) above v cong . Then greater flow rate qon°nt) is allowed via light signal operation. If under this greater flow rate q|,™nt) the onset of congestion occurs at the bottleneck once more, an incipient congested pattern begins to propagate upstream. As a result, the speed at the detector for feedback control decreases: The condition v(det) < vcong is satisfied once more. This leads to a new decrease in the flow rate qOn°nt>, and so on. To compare ALINEA with ANCONA, we chose the flow rates qin and qon related to an onramp bottleneck at which an GP appears at the bottleneck when there is no on-ramp inflow control. We find that an appropriate choice of the optimal occupancy o ^ ' in the ALINEA method leads to a suppression of GP emergence due to a decrease in the flow rate (29) in comparison with qon using light signal operation in the on-ramp lane (Fig. 7(a)). At the same
198 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction traffic demand in ANCONA, an LSP remains at the bottleneck (Fig. 7(b)), i.e., congestion occurs that is localized at the bottleneck. Nevertheless, because the speed within the LSP is not very low (about 80 km/h) the difference between the travel times on the main road in these methods cannot almost be distinguished (Fig. 7(c)). In contrast, in ALINEA the waiting time at the light signal (curve 2) is considerably longer than in ANCONA (curve 1 in Fig. 7(d)). Simulations show that the throughput in ANCONA is greater than in ALINEA. Moreover, it turns out that if short-time perturbations in the flow rate qon and/or qin occur, then ALINEA cannot prevent GP emergence at the bottleneck. This is associated with the metastability of free flow with respect to congested pattern formation, which is realized between the boundaries FS(B) and Ft(hB) in the diagram of congested patterns (Fig. 4(a)). Strictly speaking, ALINEA can maintain reliably free flow at the bottleneck if the flow rate downstream of the bottleneck is less than minimum freeway capacities of free flow at the bottleneck. This is because only in this case the probability for speed breakdown is zero, hi ANCONA, congestion is allowed to set in at the bottleneck. Therefore, greater throughputs are possible. In contrast with ALINEA, in ANCONA perturbations in the flow rate qon and/or qm cannot cause GP emergence at the bottleneck.
Fig. 7. Comparison of the ALINEA and ANCONA methods, (a, b) Vehicle speed on the main road in space and time for the ALINEA method (a) and for the ANCONA method (b). (c, d) Dependence of the travel time on the main road (c) and in the on-ramp lane (d) on time. In (c, d) curves 1 for the ANCONA method and curves 2 for the ALINEA method are shown. Thus, benefits of the ANCONA method in comparison with the ALINEA method are: 1. Achievement of greater throughput under very high traffic demand when congestion has to occur somewhere in a traffic network. In the latter case, a more homogeneous distribution of congestion among bottlenecks in the network may be possible via the application of ANCONA.
Microscopic three-phase traffic theory and applications 199 2. Considerably shorter vehicle waiting times at the light signal in the on-ramp lane(s). 3. Upstream propagation of congestion on the main road does not occur regardless of short-time perturbations in traffic flow and even if a congested pattern occurs at the bottleneck: In ANCONA, the congested pattern is spatially localized at the bottleneck. EFFECT OF AUTOMATIC CRUISE CONTROL ON CONGESTED PATTERNS AT ON-RAMP BOTTLENECK In this section, based on numerical simulations of the model considered above we discuss an influence of automatic cruise control (ACC) vehicles on traffic flow at an on-ramp bottleneck (see references in Kerner (2004)). ACC is one of the important ways of enhancing driver comfort and safety in freeway traffic (e.g., Becker et al, 1994; van Arem et al, 1996). An ACC vehicle measures the space gap gn = xln - x n - d and the relative speed vgn - v n . The ACC vehicle calculates the current time gap between the ACC vehicle and the preceding vehicle. In real ACC systems, two or more ranges of speed can be chosen where different dynamic rules are used. For simplicity we will discuss a hypothetical ACC system in which there is only one dynamic rule for the ACC vehicle in the whole possible range of vehicle speed. In most known ACC systems, at least in one of the speed ranges the dynamic behavior of the ACC vehicle can be approximately described by the well-known equation (e.g., Becker et al, 1994; van Arem et al, 1996; McDonald and Wu, 1997): a =K 1 (g n -vnT) + K 2 (v, >n - v n ) ,
(31)
where aJ,ACC' is ACC vehicle acceleration; tj A C C ) is a desired time gap of the ACC vehicle; K, and K2 are coefficients of ACC adaptation. In simulations of ACC vehicle influence on congested patterns at the bottleneck, there are vehicles that have no ACC system and ACC vehicles. Vehicles that have no ACC system move in accordance with the model described above. The ACC vehicles are randomly distributed on the road between other vehicles that have no ACC system. The ACC vehicles move in accordance with Eq. (31) where, in addition, some limitations of ACC vehicle acceleration and deceleration as well as the safe speed are used to avoid collisions. Automatic cruise control with quick dynamic adaptation To study the influence of ACC vehicles on congested patterns, we assume that at chosen qin and qon an GP occurs at the bottleneck, if there are no ACC vehicles (Fig. 8(a)). ACC vehicles can decrease the amplitude of moving jams in the initial GP if the coefficients of ACC adaptation Kj and K 2 are large enough. In this case, the ACC vehicle quickly reacts to changes in time gap and speed difference to the preceding vehicle. Moving jam suppression in the initial GP due to ACC vehicles can be seen in Figs. 8(b-d). Here, at the same qon and qin as those in Fig. 8(a) the percentage of ACC vehicles, y, in the flow rates qon and qin
200 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction increases (Figs. 8(b-d)). There is some critical y = y cr . When this critical percentage of ACC vehicles is reached, there are almost no moving jams in the congested pattern on the main road upstream of the bottleneck. If y further increases, no new moving jams occur in the congested pattern. Thus, ACC vehicles can prevent moving jam emergence.
50 100 ACC percentage Fig. 8. Influence of ACC vehicles with quick dynamic adaptation on an GP at an on-ramp bottleneck, (a-d) - Speed on the main road at different y. (e, f, g) - Discharge flow rate (e), the flow rate within the congested pattern (f) and travel time (g) as functions of y. TjACC) = 1 s, ycr = 39% , K, = K| 0 ) /max(v n ,v m m ), K y cr , then ACC vehicles induce the onset of congestion at the bottleneck (Fig. 9(b)). As a result, an GP appears at the bottleneck. The greater y, the greater the frequency of moving jam emergence in the GP (Figs. 9(c, d)). CONCLUSIONS 1. Three-phase traffic theory can explain and predict all known reproducible empirical features of phase transitions and congested patterns at freeway bottlenecks. 2. An F—>S transition governs phase transition phenomena in free flow at bottlenecks. Wide moving jams can occur spontaneously only in synchronized flow.
202 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 3. There is a deep connection between the diagram of congested patterns at a bottleneck with freeway capacity for free flow at the bottleneck: The threshold boundary for an F—>S transition in this diagram determines an infinity of the minimum freeway capacities, whereas the boundary at which the F -» S transition occurs spontaneously during a given time interval with the probability, which is equal to 1, determines an infinity of the maximum freeway capacities. 4. There are metastable regions in the diagram of congested patterns at the bottleneck in which different types of patterns can coexist. In other words, at the same traffic demand depending on perturbations in traffic flow various patterns can occur in an initial free flow at the bottleneck. However, the lifetime of these patterns, which is associated with pattern stability, can be qualitatively different. 5. If two adjacent bottlenecks are close to one another, complex patterns and nonlinear pattern interaction effects are realized. In particular, congestion at an upstream bottleneck can intensify congestion at a downstream bottleneck. 6. Different kinds of freeway traffic control methods and vehicle assistance systems can be investigated in simulations based on three-phase traffic theory concerning their influence on the efficiency of control systems, as well as comfortable and safe driving. This is because a microscopic simulation environment based on three-phase traffic theory is in accordance with empirical spatiotemporal features of congested patterns. Thus, this is a high-value instrument in a study of different freeway traffic control strategies and traffic assistance systems before they are introduced to the market. ACKNOWLEDGEMENTS I thank Sergey Klenov for his help. REFERENCES Becker, S., M. Bork, H. T. Dorissen, G. Geduld, O. Hofmann, K. Naab, G. Nocker, P. Rieth, J. Sonntag (1994). Summary of Experience with Autonomous Intelligent Cruise Control (AICC). In: Proceedings of the 1st World Congress on Applications of Transport Telematics and Intelligent Vehicle-Highway Systems, pp. 1828-1843. Elefteriadou, L., R. P. Roess and W. R. McShane (1995). Probabilistic nature of breakdown at freeway merge junctions. Transportation Research Record 1484, 80-89. Davis L.S. (2004). Multilane simulations of traffic phases. Phys. Rev. E, 69, 016108. Gartner, N. H., C. J. Messer and A. Rathi (eds.) (1997). Special Report 165: Revised Monograph on Traffic Flow Theory (Transportation Research Board, Washington, D.C.). Gipps, P. G. (1981) A behavioral car-following model for computer simulation. Trans. Res. B. 15,105-111. Hall, F. L., V. F. Hurdle and J. H. Banks (1992). Synthesis of Recent Work on the Nature of Speed-Flow and Flow-Occupancy (or Density) Relationships on Freeways. Transportation Research Record 1365, 12-18.
Microscopic three-phase traffic theory and applications 203 Helbing, D. (2001). Traffic and Related Self-Driven Many-Particle Systems. Rev. Mod. Phys. 73,1067-1141. Helbing, D., A. Hennecke and M. Treiber (1999). Phase diagram of traffic states in the presence of inhomogeneities. Phys. Rev. Lett., 82, 4360-4363. Jiang R. and Wu, Q.-S. (2004). Spatial-temporal patterns at an isolated on-ramp in a new cellular automata model based on three-phase traffic theory. /. Phys. A, 37, 81978213. Kerner, B. S. (1998). Experimental Features of Self-Organization in Traffic Flow. Phys. Rev. Lett, 81, 3797-3800. Kerner, B. S. (2002). Empirical macroscopic features of spatial-temporal traffic patterns at highway bottlenecks. Phys. Rev. E, 65, 046138. Kerner, B. S. (2004). The Physics of Traffic (Springer, Berlin, New-York, Tokyo). Kerner, B. S. and S. L. Klenov (2002). A microscopic model for phase transitions in traffic flow. J. Phys. A: Math. Gen., 35, L31-L43. Kerner, B. S. and S. L. Klenov (2003). A microscopic theory of spatial-temporal congested traffic patterns at highway bottlenecks. Phys. Rev. E, 68, 036130. Kerner, B. S. and S. L. Klenov (2004). Probabilistic Breakdown Phenomenon at On-Ramp Bottlenecks in Three-Phase Traffic Theory. Cond-mat/0502281. E-print in http://arxiv.org/abs/cond-mat/0502281. Kerner, B. S., P. Konhauser and M. Schilke (1995). Deterministic spontaneous appearance of traffic jams in slightly inhomogeneous traffic flow. Phys. Rev. E, 51, 6243-6246. KrauB, S., P. Wagner and C. Gawron (1997). Metastable states in a microscopic model of traffic flow. Phys. Rev. E, 53, 5597-5602. Lee, H. Y., H.-W. Lee and D. Kim (1998). Origin of synchronized traffic flow on highways and its dynamic phase transitions. Phys. Rev. Lett., 81, 1130-1133. Lee, H. K., R. Barlovic, M. Schreckenberg and D. Kim (2004). Mechanical restriction versus human overreaction triggering congested traffic states. Phys. Rev. Lett., 92, 238702. Lorenz, M., L. Elefteriadou (2000). A probabilistic approach to defining freeway capacity and breakdown. Transportation Research Circular E-C018, 84-95. Persaud, B., S. Yagar and R. Brownlee (1998). Exploration of the Breakdown Phenomenon in Freeway Traffic, Transportation Research Record, 1634, 64-69. McDonald, M. and J. Wu (1997). The Integrated Impacts of Autonomous Intelligent Cruise Control on Motorway Traffic Flow. In: Proceedings of the LSC 97 Conference. (Boston, USA 1997). Papageorgiou, M., H. Hadj-Salem and J.-M. Blosseville (1991). ALINEA: a local feedback control law for on-ramp metering. Transportation Research Record, 1320, 58-64. Papageorgiou M., and A. Kotsialos (2002). Freeway ramp metering: An overview. IEEE Transactions on ITS 3, 271-281. van Arem, B., J. H. Hogema and S. A. Smulders (1996). The Impact of Autonomous Intelligent Cruise Control on Traffic flow. In: Proceedings of 3rd World Congress on Intelligent Transport Systems.
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11
A BEHAVIOURAL APPROACH TO INSTABILITY, STOP AND GO WAVES, WIDE JAMS AND CAPACITY DROP Chris Tampere, Department of Traffic and Infrastructure Katholieke Universiteit Leuven, Belgium; Serge Hoogendoorn, Delft University of Technology, Faculty of Civil Engineering and Geoscience; and Bart van Arem, TNO Inro, Department of Traffic and Transport and University ofTwente, Applications of Integrated Driver Assistance (AIDA)
INTRODUCTION Congested traffic flow dynamics have attracted much interest over the last decade. To explain congested traffic flow phenomena, some researchers have stressed the importance and validity of first order hydrodynamic models, while others point out the importance of second order phenomena (instability) for the formation of various congestion patterns, and of local perturbations initiating phase transitions. In this paper, a macroscopic traffic flow model is used to examine typical congestion related phenomena like traffic flow instability, stop-and-go waves, wide jams, hysteresis, and capacity drop, and the role of driver behaviour therein. The model relates individual carfollowing behaviour to macroscopic traffic flow dynamics, and accounts for the principal aspects of individual longitudinal behaviour like: finite reaction times, anticipation behaviour to conditions downstream (i.e. non-locality and anisotropy), and finite, speed dependent space requirements of drivers. Because of the strong relation with individual human driving behaviour and the gas-kinetic mathematics that underlie this model, it is called the humankinetic traffic flow model. Moreover, the parameters that govern the individual driver behaviour in the human-kinetic model are not necessary constant in time, but may vary with traffic flow conditions, location, weather, and other conditions. In this sense, the humankinetic traffic flow model is an experimental framework in which the role of variable individual longitudinal driver behaviour in macroscopic traffic flow dynamics can be examined. This model is used to examine why different congested traffic patterns that have been reported in the literature can emerge and under what conditions. The stability of the
206 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction deceleration process in the upstream jam front turns out to be of crucial importance as well as the queue discharge rate of the accelerating jam front downstream. To increase the insight into the nature of these macroscopic traffic flow properties, the human-kinetic model is used to identify the individual behavioural parameters having dominant influence. Furthermore, it is unlikely that these behavioural parameters remain constant in time. Therefore, the effect of some plausible assumptions regarding the variability of these parameters as a function of traffic conditions is analysed. Literature review Congestion formation was already accounted for in the earliest traffic flow models by Lighthill & Whitham (1955) and Richards (1956). Ever since, numerous authors have proposed more refined traffic flow models, motivated by the fact that some types of congested traffic, like for example stop and go waves, are not adequately described by LWR (or first order) theory. A well-known example is the second order model proposed by Payne (1971). In the second half of the 1990'.? the theme of congested traffic dynamics gained renewed interest when German physicists published empirical observations and theory of a previously unknown variety of congested patterns. Kerner and co-workers claimed the discovery of 'wide moving jams' (Kerner & Rehborn, 1996a) and of 'synchronised traffic' (Kerner & Rehborn, 1996b). The author synthesised and formalised a series of publications in his 'three phase theory' (Kerner, 1999), and provided simulation models corresponding to this theory (Kerner & Klenov, 2002). Helbing et al. (1999), almost simultaneously with Lee et al. (1999), published a theory based on simulation results and later empirical confirmation (Treiber et al., 2000; Lee et al., 2000), herewith broadening the spectrum of congested traffic patterns by discerning 'pinned' and 'localised clusters', 'triggered stop and go waves', 'oscillating congested traffic', and 'homogeneous congested traffic'. Empirical phenomena like traffic hysteresis, discovered as early as 1974 by Treiterer & Myers, were analysed and included in LWR and Payne-type models, for instance by Newell (1965) and Zhang (1998, 1999). Capacity drop, or the 'two-capacity phenomenon' (Banks, 1991) was introduced in LWR theory by Lebacque et al. (1998) by considering finite acceleration capability and by Daganzo (2002) via an adaptation of the fundamental diagram. The current article aims at providing insight into the key phenomena governing the aforementioned congested traffic flow patterns: traffic instability and the outflow or queue discharge from a jam. As such, the work presented in this article follows a tradition of theoretical in-depth analyses, like the work of Zhang (1999) on traffic hysteresis, that of Herrmann & Kerner (1998) on local cluster effects, and the critical review of macroscopic traffic flow models by Lebacque & Lesort (1999). Structure of the article The remainder of this article is structured as follows. The article starts with the definition of the traffic scenario that is considered in the simulations throughout the article and an overview of the assumptions on longitudinal individual driver behaviour, hi the following section, these behavioural assumptions are formalised, defining the macroscopic humankinetic traffic flow model. Then, we illustrate various congested traffic patterns that can be obtained with this model, which leads to the central questions of this article, inquiring into the
Instability, stop and go waves, wide jams and capacity drop 207 nature of these patterns, the conditions in which they occur, and the role of individual driver behaviour therein. The next three sections are then devoted to answering these questions, with the human-kinetic traffic flow model providing guidance in these analyses. First, we analyse the different congestion patterns at the macroscopic level, and highlight the role of traffic flow stability and the queue discharge rate. The next section dives deeper into these crucial aspects of traffic flow dynamics with the aim of gaining insight into the role of individual driver behaviour in these phenomena. We then elaborate on the effect of temporary variations of individual driver behaviour and their impact on traffic flow dynamics. Finally, the article concludes with a discussion of the main findings and an outlook to further research.
SPECIFICATION OF TRAFFIC SCENARIOS AND DRIVER BEHAVIOUR The scope of this paper is limited to the influence of longitudinal driver behaviour on traffic flow dynamics on motorways. A traffic scenario is analysed of a single motorway lane without overtaking. Congestion is triggered either by considering a jam in the initial conditions (without specifying any cause for the initial jam) or by considering merging traffic at an on-ramp. In all cases, the boundary condition upstream1 is a constant flow in equilibrium (no time-varying flows at neither on-ramp nor main lane) and open downstream (Figure 1).
Figure 1
Simulated scenario
In the model and analyses considered in this paper, the following assumptions hold for the longitudinal behaviour of drivers on the motorway: all drivers behave according to the same set of behavioural rules; every individual driver has the same parameters as all other members of this class, except for the desired speed that is drawn from a normal distribution (Hoogendoorn, 1999; Helbing 1997); a driver is always in one of two modes: free flowing (unconstrained) or car-following (constrained); in either case the control behaviour is characterised as a delayed stimulus-response process with: o apure time delay equal to the reaction time of the driver;
1 Note that this assumption does not hold when a queue spills back into this boundary. Solving this issue is not trivial in case of the non-local human-kinetic model. In the simulations in this article we have circumvented this issue by extending the spatial domain upstream if necessary and plotting only the relevant x-domain.
208 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction o
the response is a choice of acceleration; the vehicle however has limited acceleration and deceleration capability, so that the desired acceleration is bounded between a minimal and maximal value; o the stimulus of an unconstrained driver is the deviation of the individual speed with respect to the desired speed; o a constrained driver has two stimuli: the (perceived or anticipated) relative speed with the predecessor and the deviation of the gap to the predecessor with respect to a speed-dependent desired gap; the parameters of the stimulus response models may depend on the actual level of activation of the driver; this is the internal state of the driver that is normally equal to some comfortable or normal level, but can be raised or decreased due to more or less demanding traffic conditions experienced recently. More specifically, these behavioural assumptions are formalised for a single individual driver as follows. With the notation v for the speed and s for the gap, the actual acceleration v at time M-Twith 2" the reaction time of a driver j with a predecessor j - \ equals:
V{t+T>J^M
;
',(0-^M')) + y,.,(0-v,(0l
(1)
accmin < v < o,ccmax
with:
(2)
and parameters: : desired speed of driver7 Wj TW, TS, TV : relaxation factor for the desired speed, gap and relative speed : minimal and maximal acceleration capability of the accmin, accmax vehicle 5,rf, s* : linear and quadratic factors for the desired gap With the notation a for the activation level, the change of a in time is given by:
da _aL»al(v)-a dt ra
}
This equation expresses that the activation level a relaxes towards the speed-dependent normal level avmrnlal (v) with a relaxation factor za. To conclude the specification of individual longitudinal driver behaviour, a specification is needed for the dependency of the longitudinal behaviour of equations (1) and (2) on the activation level a. Little theoretical or experimental knowledge is available to date to validate any specification in this respect. In order to illustrate the potential influence of the activation level on driver behaviour, the plausible assumptions will be made further in this paper that less active driving can be identified either with more moderate levels of acceleration or with
Instability, stop and go waves, wide jams and capacity drop 209 larger values for the desired gaps (see section on variable driver behaviour). In the scenarios with on-ramps, no specific behavioural rules are specified with respect to individual merging behaviour, other than the following assumptions: the exact position along the on-ramp of merging into the main lane is drawn from a triangle-shaped probability distribution; prior to merging onto the main lane, drivers first adapt their speed on the on-ramp to the average speed of the main lane flow at the point of merging; drivers merging into the main lane have an activation level that is equal to the average activation level of drivers in the main lane flow at the point of merging.
FORMALISATION OF BEHAVIOURAL SPECIFICATIONS: THE HUMAN-KINETIC MODEL The human-kinetic traffic flow model The behavioural assumptions of the previous section have been formalised in a kinetic traffic flow model, according to the theory originally proposed by Prigogine & Herman (1971). The core of this model is the formulation of a generalised conservation law for vehicles with certain commonalities in their individual states S'. The conservation law expresses the conservation of the probability density p(t,S'), which is defined so that the expected number of vehicles having state »S"e[5",5"+dS") at time t equals p(t,S') dS" (Hoogendoorn ,1999; Leutzbach, 1988):
a
* r dt) { dt ) ,
M
In this equation, the dependency of p o n (t,S') has been omitted for notational simplicity; the product operator denotes the inner product and Vs, the Nabla operator for the state vector 5" with dimensions (x,si,S2, ...,sn). The method of moments is traditionally used2 to transform the kinetic continuity equation (4) into its equivalent macroscopic formulation (Helbing, 1997). With this method, dynamic equations for the ii order moment of the probability density function p for state variable s, are obtained by multiplication of both RHS and LHS of equation (4) by s" and consequent integration over all state variables in S'. It can be shown that with S'=(x,v) and for values of K equal to 0 and 1 respectively the following macroscopic 'momentum' equations are obtained:
It should be mentioned here that an alternative procedure exists, based on a Chapman-Enskog expansion of the kinetic traffic model (Nelson & Sopasakis, 1999). Although this approach has some elegant theoretical advantages, little is known about the practical applicability of the resulting sequence of traffic flow models, which was the motivation to use the more traditional, but more ad hoc approach of the method of moments and associated higher-order models.
210 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction
»
+
8t dkV2
8kV +
8t
8x
dk® +
=
^x_
J^
convec ion
pressure
(5)
^ =W \dt)iiscrete , I dv\ k{—
Xdtj^ smootn
acceleration
f
+
(dp) v ——
J»
dv
(6)
\.dt)dhcrae
discrete
acceleration
In these equations, the following symbols and notations are used: k : macroscopic density : macroscopic or average speed V © : speed variance (the product k® is often indicated as the traffic pressure) tne {y(x))x expected value of a function y(x) with respect to all possible values of x Equation (5) is the macroscopic continuity equation for the vehicular density k. The term in the RHS stands for flow of vehicles into or out off the road section under
— ^
d t
) discrete
consideration. Equation (6) describes the dynamics of the flow rate Q=kV, also called the speed momentum equation or conservative speed equation. Note that there are two entries in equation (6) in which changes of the speed can be considered: smooth acceleration: change of the average speed V due to smooth individual accelerations; discrete acceleration: change of the average speed V due to any event causing a discrete change of p{tjc,v), the expected number of vehicles with speed v; The human-kinetic model of this paper uses this generic approach with two specific characteristics: acceleration and deceleration of vehicles in the flow are considered simultaneously and as smooth processes in the so-called acceleration integral, based on an individual longitudinal behavioural model; the state 5" of individual vehicle-driver combinations is extended with the internal state of the driver, the activation level a, in addition to the position and longitudinal speed: S'=(x,v,a). The characteristic specifications of the human-kinetic model are summarised in the next subsections. For a more elaborate description of the model, its specifications and properties the reader is referred to Tampere (2004). The acceleration integral The acceleration in the speed momentum equation (6) is generally determined by the specification of the smooth and discrete acceleration terms. Differently from existing kinetic traffic flow models, where deceleration interactions of faster with slower vehicles are
Instability, stop and go waves, wide jams and capacity drop 211 modelled as instantaneous events via the discrete acceleration term, no discrete changes of the speed (momentum) are considered in the human-kinetic traffic flow theory (except for contributions to the total speed momentum by in- or outflow of traffic near on- and offramps). The gross acceleration of the flow is accounted for in the smooth acceleration term, and calculated as a macroscopic resultant of individual acceleration or deceleration decisions, according to the individual longitudinal model presented in equations (1) and (2):
w
s
L^^J) ) =
